Scholium: Sectoral Probability, Measurement, and Dual Information Flow from the Law of Conservation of Probability in the Theory of Entropicity (ToE)

The probability law of the Theory of Entropicity (ToE) is frequently misunderstood when interpreted through the lens of classical or Copenhagen‑style measurement theory. In ToE, the relation

$$P_o(t)+P_e(t)=1$$

does not refer to what a human observer sees, nor does it presuppose the presence of a conscious agent. Instead, it expresses a sectoral decomposition of the total Hilbert space, reflecting how the universe partitions amplitude between two orthogonal components:

• the coherent (observer) sector ${\mathcal{H}}_o$, and

• the entropic sector ${\mathcal{H}}_e$.

This decomposition is encoded in the structural relations

$${\psi }_o(t)\perp {\psi }_e(t),{\mathcal{H}}_{\mathrm{t}\mathrm{o}\mathrm{t}}={\mathcal{H}}_o\oplus {\mathcal{H}}_e,\mathrm{\parallel }\mathrm{\Psi }(t){\mathrm{\parallel }}^2=\mathrm{\parallel }{\psi }_o(t){\mathrm{\parallel }}^2+\mathrm{\parallel }{\psi }_e(t){\mathrm{\parallel }}^2\mathrm{.}$$

These are statements of geometry, not psychology.

1. Measurement in ToE is not human‑dependent

ToE explicitly rejects the Copenhagen claim that physical reality depends on human observation. It does not require consciousness, perception, or an experimenter to bring phenomena into existence. The Moon exists whether or not anyone looks at it. Measurement, in ToE, is an entropic process, not a mental act.

Thus, ToE is fully consistent with an observer‑independent external world.

2. Measurement is “observer‑dependent” only in a technical, sectoral sense

When ToE refers to an “observer,” it does not mean a person. It means the coherent sector ${\mathcal{H}}_o$ of the Hilbert space: the subspace capable of supporting stable, classical records. This sector is defined by:

• coherence,

• information accessibility,

• low entropy, and

• the ability to retain classical information.

“Observer‑dependent” therefore means:

dependent on which degrees of freedom remain coherent enough to register information.

It does not mean dependent on a human presence.

3. The entropic sector is the complementary domain

The entropic sector ${\mathcal{H}}_e$ is characterized by:

• increasing entropy,

• loss of coherence,

• dynamical irreversibility, and

• inaccessibility of fine‑grained quantum information.

This is the sector into which microscopic details dissipate under the entropic evolution operator $e^{-Ct}$.

4. The probability law expresses sectoral conservation, not subjective observation

The relation

$$P_o(t)+P_e(t)=1$$

is a conservation law describing how amplitude flows between ${\mathcal{H}}_o$ and ${\mathcal{H}}_e$. It is not a statement about what a person sees. It is a structural identity arising from the orthogonal decomposition of the total state.

Thus, the ToE probability law is sectoral, not psychological.

5. Why ToE calls measurement “observer‑dependent”

Measurement in ToE is the projection of the total state onto the coherent sector:

$$\mathrm{\Psi }(t)\mapsto {\psi }_o(t)\mathrm{.}$$

This projection depends on:

• which degrees of freedom remain coherent,

• which have decohered,

• which are accessible to ${\mathcal{H}}_o$, and

• which have been entropically suppressed into ${\mathcal{H}}_e$.

This is analogous to:

• simultaneity in relativity,

• electric vs. magnetic field components,

• kinetic vs. potential energy.

All are frame‑dependent, not human‑dependent.

6. The consistency of ToE’s position

ToE therefore asserts:

• The Moon exists without a human observer. Measurement is determined by entropic thresholds, not consciousness.

• Measurement is observer‑dependent because the coherent sector is defined by the physical structure of the system.

• Probability is conserved across sectors

$$P_o(t)+P_e(t)=1.$$

• The partition is relative, but the total is invariant.

There is no contradiction—only a precise distinction between physical sectors and human observers.

7. Dual information flow: classical accessibility vs quantum inaccessibility

The apparent tension between “information becomes measurable” and “information becomes inaccessible” dissolves once we distinguish two kinds of information:

Classical information (accessible to ${\mathcal{H}}_o$)

• macroscopic

• coarse‑grained

• stable

• measurable

Quantum micro‑information (lost to ${\mathcal{H}}_o$)

• fine‑grained

• phase‑sensitive

• coherence‑dependent

• absorbed by ${\mathcal{H}}_e$

Thus, when a system crosses the entropic threshold:

• classical information becomes accessible (birth of a classical record),

• quantum information becomes inaccessible (loss of coherence).

These are not contradictory; they are two sides of the same entropic flow.

8. Conservation unifies the two flows

The conservation law

$$P_o(t)+P_e(t)=1$$

expresses that:

• the observer sector gains classical probability,

• the entropic sector gains lost quantum probability,

• the total remains conserved.

Measurement is therefore the transfer of coherence into entropy, producing classical information while dissipating quantum microstructure.

9. The ToE declaration

ToE states:

Measurement makes classical information accessible, while quantum information becomes inaccessible.

Both statements are true. They describe different layers of the same entropic process.

Entropic Probability Conservation and the Decomposition Po(t)+Pe(t)=1

A central structural feature of the Theory of Entropicity (ToE) is the division of physical evolution into two orthogonal sectors: the observer (coherent) sector and the entropic sector. This division is not merely conceptual; it is encoded directly in the Hilbert‑space architecture of the theory and leads to a distinct probability‑conservation law that differs from the classical Kolmogorov formulation.

To formalize this structure, the total Hilbert space is decomposed as

$${\mathcal{H}}_{\mathrm{t}\mathrm{o}\mathrm{t}}={\mathcal{H}}_o\oplus {\mathcal{H}}_e,$$

where ${\mathcal{H}}_o$ represents the coherent observer sector and ${\mathcal{H}}_e$ represents the entropic sector. The ToE evolution operator acts on the total state through a combined unitary–entropic flow,

$$U_{\mathrm{T}\mathrm{o}\mathrm{E}}(t)=e^{-iHt}{e}^{-Ct},$$

where $H$ generates coherent evolution and $C$ generates entropic dissipation. Under this evolution, the total state decomposes as

$$\mathrm{\Psi }(t)={\psi }_o(t)+{\psi }_e(t),$$

with the orthogonality condition

$${\psi }_o(t)\perp {\psi }_e(t)\mathrm{.}$$

Norm conservation of the total state,

$$\mathrm{\parallel }\mathrm{\Psi }(t){\mathrm{\parallel }}^2=1,$$

implies the additive relation

$$\mathrm{\parallel }\mathrm{\Psi }(t){\mathrm{\parallel }}^2=\mathrm{\parallel }{\psi }_o(t){\mathrm{\parallel }}^2+\mathrm{\parallel }{\psi }_e(t){\mathrm{\parallel }}^2\mathrm{.}$$

Defining the sectoral probabilities as

$$P_o(t):=\mathrm{\parallel }{\psi }_o(t){\mathrm{\parallel }}^2,P_e(t):=\mathrm{\parallel }{\psi }_e(t){\mathrm{\parallel }}^2,$$

one obtains the entropic probability‑conservation law,

$$P_o(t)+P_e(t)=1.$$

This relation is not a restatement of the classical normalization axiom ${\sum }_i{P}_i=1$. Instead, it expresses a binary partition of the total quantum state into two dynamically coupled but orthogonal sectors. Classical probability theory partitions events; the Theory of Entropicity partitions sectors of physical reality. The observer sector captures coherent, information‑bearing evolution, while the entropic sector captures the irreversible flow of amplitude into the informationally inaccessible domain generated by $C$.

Thus, the equation

$$P_o(t)+P_e(t)=1$$

is a conservation law arising from the Hilbert‑space structure of ToE and the combined unitary–entropic dynamics. It encodes the fundamental principle that while amplitude may flow from the observer sector into the entropic sector, the total probability remains conserved across the full ToE evolution. This decomposition provides the mathematical foundation for entropic irreversibility, observer‑dependent coherence, and the emergence of classicality within the ToE framework.

Probability as a Conservation Law in the Theory of Entropicity (ToE)

One of the most striking conceptual departures introduced by the Theory of Entropicity (Toe) is the re‑interpretation of probability itself. In classical physics and in the Kolmogorov framework, probability is defined axiomatically: the sum of all mutually exclusive outcomes must equal unity. This rule has no dynamical origin; it is not derived from physical principles, nor does it arise from the geometry of the underlying state space. It is simply imposed.

In contrast, ToE does not assume probability conservation. It derives it.

The starting point is the structural decomposition of the total Hilbert space into two orthogonal sectors,

$${\mathcal{H}}_{\mathrm{t}\mathrm{o}\mathrm{t}}={\mathcal{H}}_o\oplus {\mathcal{H}}_e,$$

where ${\mathcal{H}}_o$ represents the coherent observer sector and ${\mathcal{H}}_e$ represents the entropic sector. Under the combined unitary–entropic evolution generated by

$$U_{\mathrm{T}\mathrm{o}\mathrm{E}}(t)=e^{-iHt}{e}^{-Ct},$$

the total state decomposes as

$$\mathrm{\Psi }(t)={\psi }_o(t)+{\psi }_e(t),$$

with the orthogonality condition ${\psi }_o(t)\perp {\psi }_e(t)$. Norm conservation of the total state,

$$\mathrm{\parallel }\mathrm{\Psi }(t){\mathrm{\parallel }}^2=1,$$

implies the additive relation

$$\mathrm{\parallel }\mathrm{\Psi }(t){\mathrm{\parallel }}^2=\mathrm{\parallel }{\psi }_o(t){\mathrm{\parallel }}^2+\mathrm{\parallel }{\psi }_e(t){\mathrm{\parallel }}^2\mathrm{.}$$

Defining the sectoral probabilities as

$$P_o(t):=\mathrm{\parallel }{\psi }_o(t){\mathrm{\parallel }}^2,P_e(t):=\mathrm{\parallel }{\psi }_e(t){\mathrm{\parallel }}^2,$$

one obtains the entropic probability‑conservation law,

$$P_o(t)+P_e(t)=1.$$

Although this expression resembles the classical normalization rule, its meaning is fundamentally different. Classical probability partitions outcomes; ToE partitions reality. The quantities $P_o(t)$ and $P_e(t)$ are not probabilities of events but probabilities associated with two dynamically coupled, orthogonal sectors of the universe. The entropic operator $e^{-Ct}$ transfers amplitude from the observer sector into the entropic sector, generating irreversibility, decoherence, and the arrow of time. Yet the total probability is conserved across the full ToE evolution.

Thus, ToE elevates probability from an epistemic bookkeeping rule to an ontological conservation law. Probability becomes a structural invariant of the universe’s Hilbert‑space geometry and its entropic dynamics. This shift—from axiom to conservation principle—marks one of the most conceptually significant contributions of the Theory of Entropicity (ToE).

What are the Conceptual, Philosophical, and Mathematical Foundations of the Theory of Entropicity (ToE)?

The Theory of Entropicity (ToE), originated by John Onimisi Obidi in early 2025, is a radical framework in modern theoretical physics that proposes entropy as the fundamental, dynamic "ontic" field underlying reality, rather than a secondary statistical byproduct. It seeks to unify thermodynamics, quantum mechanics, and general relativity by establishing entropy as the primary causal substrate of the universe. 

Conceptual and Mathematical Foundations of ToE 

• Entropy as a Fundamental Field (Ontic Entropy): ToE flips the conventional hierarchy by promoting entropy to an ontological scalar field that permeates existence. It acts as a continuous and dynamic field that drives all physical processes.
• Emergent Spacetime and Gravity: Spacetime is not a container, but an emergent map of entropic gradients (spatial organization). Gravity is reinterpreted not as a fundamental force or just spacetime curvature, but as an emergent phenomenon caused by the field’s tendency to maximize entropy.
• The "No-Rush Theorem": Colloquially summarized as "God or Nature Cannot Be Rushed" (G/NCBR), this theorem posits that no physical interaction can occur instantaneously. All processes take a finite, non-zero time to rearrange the entropic field, providing a physical basis for causality.
• Speed of Light as an Entropic Rate: The speed of light (c) is reinterpreted as the maximum possible rate at which the entropic field can reorganize energy and information, establishing the "speed of causality".
• Obidi Action and Master Entropic Equation (MEE): The dynamics of the entropic field are governed by the Obidi Action (a variational principle), leading to the MEE—the entropic equivalent to Einstein's field equations. 

Philosophical Foundations of ToE 

• Ontodynamics: The philosophical core of ToE is Ontodynamics, defined as the study of existence as entropic motion. It investigates how phenomena, interactions, and observations evolve through entropy-driven dynamics.
• From Order to Vitality (Heraclitean Flux): ToE rejects the idea that entropy is merely decay. It reinterprets entropy as the "heartbeat of existence" and the active force of transformation that gives rise to complexity, life, and self-organization.
• Information-Geometric Ontology: Information is considered the primary "material" of reality. The theory claims that information possesses geometry, and geometry possesses dynamical agency. Information-geometric tools, such as the Amari-Čencov alpha-connection, are treated as real physical entities describing the deformation of space by entropy. In the Theory of Entropicity (ToE), entropy is what both creates and deforms what we identify as spacetime. This is Entropic Dynamics in the Theory of Entropicity (ToE).
• Iterative Universe (Computation): ToE suggests the universe is a continuous, self-correcting computation. The equations of ToE are non-explicit and iterative, mirroring how information is updated via Bayesian inference.
• Unifying Metaphor (Chronos and Pyros): ToE resurrects ancient philosophical intuitions by uniting the "Chronos" (the irreversible flow of time via entropy) with "Pyros" (the fiery, maximum rate of transformation/light). 

ToE is distinct from other entropic models (like Erik Verlinde’s) because it treats entropy as a physical, foundational field rather than just an emergent force, and is distinct from epistemic views by asserting entropy's ontic (real) nature. 

 

Communications Between Daniel Moses Alemoh and John Onimisi Obidi on the Foundations and Formulation of the Theory of Entropicity (ToE): Dialogues on a New Theory of the Foundation of Modern Theoretical Physics—Part II

Preamble 

Scientific revolutions often germinate through private correspondence rather than polished manuscripts.  Between 2025 and 2026 John Onimisi Obidi shared a developing theoretical program with Daniel Moses Alemoh.  Obidi proposed that entropy is not a derivative thermodynamic bookkeeping quantity but the primary field from which space, time, matter and information emerge.  This radical inversion of twentieth‑century physics treats entropy as a dynamical scalar field S(x) defined on an entropic manifold.  Obidi and Alemoh debated how to formalize this idea, how to reinterpret constants like the speed of light, how to explain cosmic expansion, and how to derive known physics from an entropic action.  This article reconstructs those dialogues into a structured review, placing them in the context of existing entropic theories and citing publicly available sources.  We argue that the Theory of Entropicity (ToE) represents a bold attempt to rebuild modern physics on an informational foundation comparable in ambition to Einstein’s elevation of c to a universal postulate.

encyclopedia.pub

1 Introduction

Correspondence has long nurtured scientific innovation.  Letters between Newton and Hooke, Einstein and Besso, or Bohr and Schrödinger often contained nascent ideas that later reshaped physics.  In that tradition, the exchanges between John Onimisi Obidi and Daniel Moses Alemoh trace the gestation of the Theory of Entropicity (ToE).  ToE calls for “abandoning the view of entropy as a secondary, statistical by‑product and instead elevating it to the status of a fundamental field”.  In analogy with Einstein’s decisive step of elevating the speed of light c, ToE posits a universal entropic field S with its own dynamics.  The central claim is that the geometry of space, the flow of time and the dynamics of motion are manifestations of entropy gradients rather than primitive structures.  This inversion implies that constants, interactions and even measurement emerge from the entropic field’s behaviour. (encyclopedia.pub)

Alemoh’s role was not merely receptive; he raised penetrating questions about the consistency of this framework.  In particular he asked how a theory in which spacetime is emergent could reconcile a finite light‑speed limit with the observed superluminal recession of galaxies, and how ToE could reproduce known physics.  The following sections organize the core themes of their correspondence and amplify them using published expositions of ToE and related entropic models.

2 The Entropic Field: Ontological Foundation

Classical physics begins with geometry or quantum fields as ontological primitives.  By contrast, ToE begins with a scalar entropic field defined on an entropic manifold S.  This field is continuous, differentiable and dynamically evolving.  Each point of the manifold has a real‑valued entropic density representing intrinsic “ontological density,” configurational multiplicity, geometric potential and information substrate.  The entropic field’s gradients behave like forces and determine “entropic geodesics,” while higher derivatives encode curvature‑like responses.  In effect, the entropic field is the substrate from which geometry, forces and information flow are derived. (theory-of-entropicity-toe.pages.dev)

The entropic field has both local and non‑local contributions.  Local variations determine immediate dynamics, while non‑local structure governs global coherence.  These features allow ToE to account for both short‑range interactions and large‑scale cosmological phenomena within the same framework.(theory-of-entropicity-toe.pages.dev)

3 The Obidi Action and the Master Entropic Equation

Correspondence between Obidi and Alemoh repeatedly returned to the need for a rigorous mathematical formalism.  In ToE the dynamical laws arise from the Obidi Action—an entropic analogue of the Einstein–Hilbert action.  The Obidi Action is a variational principle which encodes the dynamics of the entropy field S.  Varying this action yields the Master Entropic Equation (MEE) or Obidi Field Equations (OFE).  These equations play the role that Einstein’s field equations play in general relativity, governing how entropy gradients evolve and couple to geometry, matter and information.  From the MEE follow secondary structures: (encyclopedia.pub)

Entropic geodesics—natural paths in the entropic manifold along which systems evolve.

Entropic potential equation—a relation governing the manifestation of entropic forces.

Unlike Einstein’s equations, which admit closed‑form solutions in highly symmetric situations, the ToE field equations are generally approached via iterative methods.  This reflects the inherently probabilistic and information‑theoretic nature of entropy; solutions are successive refinements rather than static metrics.  The iterative character underscores ToE’s view that physical laws are emergent equilibria of continuous entropic computation rather than fixed constraints. (encyclopedia.pub)

4 Iterative Nature of the OFE and the Vuli–Ndlela Integral

The OFE describe the continuous evolution of the entropy field, not the curvature of a fixed spacetime.  They imply that entropy is not a passive measure but an active generative principle that reorganizes reality.  Solving the OFE requires starting from an initial informational configuration and allowing it to evolve through successive entropy updates.  Each iteration yields a more stable entropic structure, analogous to Bayesian updating.  In this perspective, the universe is an ongoing computation: it never “arrives” at a configuration but continuously recalculates its entropic state. (encyclopedia.pub)

The Vuli–Ndlela Integral generalizes Feynman’s path integral to entropy.  Instead of summing over mechanical trajectories, it sums over entropic configurations of the universe’s informational state.  Each configuration is weighted by both a causal phase and an entropic attenuation that accounts for irreversible growth or redistribution of entropy.  Information geometry provides the natural mathematical setting: probability distributions form a curved manifold whose curvature is interpreted physically as gravitational, electromagnetic and quantum phenomena.  Hence ToE unites path integrals and information geometry, expressing physical evolution as an unending dialogue between entropy, information and geometry. (encyclopedia.pub)

5 Dialogues on the Speed of Light and Cosmic Expansion

A key theme in Alemoh’s correspondence concerned the interpretation of the speed of light.  Standard physics treats as a fundamental invariant entering Lorentz symmetry.  ToE, however, interprets as the maximum rate at which the entropic field can reorganize information.  This finite rate governs causal interactions and becomes the emergent constant observed in relativity.  Thus is a property of the present entropic regime rather than an immutable number.  If the dynamics of the entropic field were different in another epoch or region, the effective value of could differ. (encyclopedia.pub)

Alemoh asked how this interpretation can coexist with super‑luminal cosmic expansion.  In standard cosmology, galaxies recede faster than because the metric expands; there is no violation of causality.  ToE explains this by distinguishing two sectors:

Local dynamical sector—internal propagation of disturbances within the entropic field.  Signals, particles and causal influences are limited by the finite entropic redistribution rate.

Global background sector—evolution of the entropic manifold itself.  Cosmic expansion is interpreted not as motion through pre‑existing space but as the growth or extension of the entropic manifold.  Hence recession speeds may exceed because the “medium” is expanding; this does not transmit information faster than.  This distinction parallels Daniel Alemoh’s analogy: light is the fastest ripple through the field, while expansion is the field itself increasing its extent (as described in Obidi’s replies).

In these dialogues Obidi emphasized that ToE must formalize this separation.  The OFE and Vuli–Ndlela Integral treat the background evolution as part of the entropic dynamics.  Locally, the finite entropy redistribution rate enforces relativity; globally, entropic growth accounts for cosmological expansion.  Thus the entropic speed limit remains intact while ToE accommodates super‑luminal recession.  External entropic theories support this view.  A mainstream news report on Ginestra Bianconi’s work notes that gravity can be derived from an entropic action coupling matter fields with geometry, underscoring that entropic actions can produce gravitational dynamics without requiring a fixed spacetime.  ToE extends this insight by providing both local and spectral Obidi Actions that yield the Master Entropic Equation, entropic geodesics and a unified description of gravity, time, quantum processes and information geometry. (popularmechanics.com), (cambridge.org)

6 Integration with External Entropic Paradigms

While ToE is original, it connects to broader efforts to derive spacetime and gravity from entropy.  Verlinde’s entropic gravity, Bianconi’s quantum relative entropy, and emergent time proposals all suggest that gravity and time may have entropic origins.  A popular exposition notes that gravity can emerge from quantum relative entropy and an entropic action.  These ideas show that entropic considerations can lead to Lorentz‑symmetric dynamics and even cosmological constants.  ToE goes beyond these frameworks by elevating entropy to a universal field and introducing the Obidi Action and Vuli–Ndlela Integral.  In this sense, ToE can be seen as unifying and extending entropic gravity programmes by providing both a local variational principle and a spectral variational principle.(popularmechanics.com), (cambridge.org)

The ToE programme also resonates with information geometry.  In information geometry the manifold of probability distributions has a natural curvature, and distances measure distinguishability.  The entropic field’s curvature in ToE plays an analogous role, linking the geometry of information to physical phenomena.  This connection suggests that the entropic manifold might correspond to the statistical manifold underlying quantum states and thermodynamic ensembles.  Thus ToE offers a conceptual bridge between physics and inference. (encyclopedia.pub)

7 Concluding Reflections and Future Work

The dialogues between Daniel Moses Alemoh and John Onimisi Obidi exemplify how critical questioning refines speculative theories.  Alemoh’s insistence on clarifying the status of c, the nature of cosmic expansion, and the formal foundations of ToE drove Obidi to sharpen his formulations.  The resulting theory is ambitious: it posits that entropy is the heartbeat of existence, not a measure of disorder; it proposes an entropic field whose gradients and curvature generate forces and geometry; it introduces an Obidi Action yielding a Master Entropic Equation analogous to Einstein’s equations; and it generalizes path integrals through the Vuli–Ndlela Integral (VNI).  These elements suggest a new foundation for physics grounded in information and irreversibility. (encyclopedia.pub), (theory-of-entropicity-toe.pages.dev)

However, ToE remains in a formative stage.  Major challenges include: deriving Lorentz symmetry and known field theories from the entropic field; computing testable predictions; understanding how quantum measurement arises; and integrating the theory with established thermodynamics.  The iterative character of the OFE implies that approximate numerical schemes will be needed.  Furthermore, philosophical questions—such as whether time becomes an emergent ordering of entropic updates—require careful analysis.

Despite these challenges, the ToE correspondence illustrates a bold ontological courage: the willingness to question entrenched primitives and to propose that reality is fundamentally informational.  If future work can bridge ToE with empirical data and established physics, the entropic field may one day stand alongside the speed of light as a new pillar of natural philosophy.

References

J. O. Obidi, Theory of Entropicity (ToE): Chapter 2 – The Entropic Field, 2025–2026, describing the entropic field as a continuous, differentiable, dynamically evolving scalar whose gradients and curvature generate forces and geometry.

theory-of-entropicity-toe.pages.dev

theory-of-entropicity-toe.pages.dev

J. O. Obidi, Theory of Entropicity (ToE): Path to Unification of Physics, Encyclopedia MDPI, 2025.  The article proposes elevating entropy to a universal field, analogous to Einstein’s elevation of c; it introduces the Obidi Action, Master Entropic Equation and entropic geodesics.(encyclopedia.pub)

J. O. Obidi, The Theory of Entropicity Goes Beyond Holographic Pseudo‑Entropy, Cambridge Open Engage, 2026.  The abstract emphasises that ToE treats entropy as the fundamental physical field equipped with local and spectral Obidi actions, producing a unified description of gravity, time, quantum processes and information geometry. (cambridge.org)

E. Rayne, “A New Theory Says Gravity May Come From Entropy—Which Could Lead to a Unified Theory of Physics,” Popular Mechanics, 20 January 2026.  The article quotes Ginestra Bianconi: “Gravity is derived from an entropic action coupling matter fields with geometry”—an external perspective supporting entropic action approaches. (popularmechanics.com)

J. O. Obidi, Theory of Entropicity (ToE): Information Geometry and the Vuli–Ndlela Integral, Encyclopedia MDPI, 2025.  Discusses how the Vuli–Ndlela Integral sums over entropic configurations weighted by causal phases and entropic attenuation, connecting ToE to path integrals and information geometry. (encyclopedia.pub)

This document synthesizes the key themes from your discussions with Daniel Moses Alemoh on the Theory of Entropicity, situating them within broader entropic and information-theoretic frameworks while retaining the conversational spirit of your exchanges. It includes citations to publicly available sources that support and expand upon the ideas explored between Daniel Moses Alemoh and John Onimisi Obidi on the foundations and formulation of the Theory of Entropicity (ToE).

Communications Between Daniel Moses Alemoh and John Onimisi Obidi on the Foundations and Formulation of the Theory of Entropicity (ToE): Dialogues on a New Theory of the Foundation of Modern Theoretical Physics—Part I (Version 2.0)

Preamble 

This paper presents a deep analytical reconstruction of the intellectual correspondence between Daniel Moses Alemoh (danielalemoh2@gmail.com) and John Onimisi Obidi (jonimisiobidi@gmail.com) concerning the conceptual architecture, mathematical aspirations, and foundational claims of the Theory of Entropicity (ToE). Far from casual exchanges, these dialogues function as a developmental workshop in which critical questions concerning the meaning of the speed of light, the emergence of spacetime, the interpretation of cosmic expansion, causality, and the role of entropy in physical ontology were repeatedly examined. The present study situates those discussions within the broader history of foundational physics, compares their themes with earlier paradigm shifts from Newtonian mechanics to relativity and quantum theory, and evaluates the internal coherence of ToE as articulated through these communications. Particular attention is given to the reinterpretation of the constant as an emergent limit of entropic redistribution, the distinction between local propagation and global manifold evolution, and the proposed formal role of the Obidi Action and Vuli Ndlela Integral. Whether ultimately validated or refuted, these exchanges constitute a serious case study in the birth of speculative theoretical physics through correspondence.

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1. Introduction: Correspondence as a Generator of Physics

Modern physics has repeatedly advanced through dialogue before publication. Einstein’s exchanges with Michele Besso preceded major conceptual clarifications in relativity. Bohr’s correspondence with Einstein refined quantum complementarity. Schrödinger’s letters sharpened wave mechanics. In each case, private questioning acted as a pre-publication stress test.

The communications between Daniel Moses Alemoh and John Onimisi Obidi belong to this intellectual tradition in form, though not yet in historical scale. They concern the Theory of Entropicity (ToE), a framework whose central thesis is radical:

Entropy is not secondary bookkeeping; entropy is primary physical reality.

This reverses the hierarchy assumed by conventional physics.

Standard physics generally treats:

• spacetime geometry,
• fields,
• particles,
• symmetry principles,

as fundamental, while entropy appears statistically or thermodynamically at higher levels.

ToE proposes the opposite order:

• entropy field first,
• geometry second,
• matter as stabilized entropic structure,
• time as irreversible entropic sequencing,
• constants as regime-properties of the field.

Daniel Alemoh’s role in the correspondence was especially important because he did not merely receive these claims; he interrogated their consistency.

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2. Methodological Scope of This Paper

This study reconstructs the themes of the correspondence from the documented exchanges and synthesizes them into formal theoretical categories:

1. Ontology of the entropic field
2. Reinterpretation of the speed of light
3. Emergence of spacetime structure
4. Cosmological expansion under ToE
5. Role of action principles
6. Entropy-weighted path selection
7. Comparative significance to existing physics

The aim is not hagiography, but disciplined exposition.

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3. Core Foundational Thesis of ToE

The recurring position communicated by Obidi is that entropy should be elevated from a derived quantity to a field variable , defined locally over reality.

Instead of entropy being computed from states, states are computed from entropy configurations.

Symbolically:

\text{Standard View: } \text{State} \rightarrow \text{Entropy}

\text{ToE View: } \text{Entropy Field} \rightarrow \text{State, Geometry, Dynamics}

This inversion has profound consequences.

If entropy is local and dynamical, then gradients, flows, thresholds, and capacities of entropy become candidates for explaining:

• motion,
• force,
• measurement,
• temporal direction,
• curvature,
• limits of propagation.

This is the conceptual backbone of the correspondence.

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4. Daniel Alemoh’s Central Contribution: The Question of

Among the most sophisticated themes in the dialogue was Daniel Alemoh’s treatment of the speed of light.

He correctly identified that ToE does not necessarily regard as primitive. Rather, within the framework:

c = \text{maximum current rate of entropic redistribution}

That is, becomes the maximal rate at which correlations, constraints, energy, or distinguishability can propagate through the entropic substrate.

This differs sharply from Einsteinian orthodoxy, where is embedded fundamentally in Lorentz symmetry.

Daniel then pressed the decisive question:

If space emerges from the entropic field, what does cosmic expansion mean when recession exceeds ?

This question is technically deep because it probes whether ToE confuses:

• speed through space, and
• evolution of space itself.

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5. The Two-Layer Resolution: Propagation vs Background Evolution

The most coherent reply developed through the exchanges is that ToE requires two dynamical layers.

5.1 Layer I: Internal Propagation

This includes:

• photons,
• particles,
• causal signals,
• local forces,
• measurement chains.

These processes are bounded by:

v \leq c

where is the entropic transfer ceiling.

5.2 Layer II: Background Manifold Evolution

This includes:

• cosmological scaling,
• entropy vacuum restructuring,
• relational node growth,
• topological re-indexing of emergent space.

These are not signal transmissions through space. They are changes in the field architecture from which space is inferred.

Hence superluminal recession need not violate the local bound.

This parallels standard cosmology formally, but differs ontologically:

• Standard view: metric expands
• ToE view: entropic relational manifold updates

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6. Daniel’s Ripple Analogy and Its Importance

Daniel described light as the fastest ripple in the field, while expansion is the field itself increasing in extent.

This analogy is stronger than it first appears.

Let:

• = propagating mode
• = medium/manifold state

Then standard propagation studies:

\partial_t u = \mathcal{D}[u;M]

But cosmic evolution concerns:

\partial_t M = \mathcal{F}(M,S)

Daniel intuitively separated the equation of disturbance from the equation of substrate.

That distinction is mathematically mature.

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7. The Variable Meaning of Constants

A recurring ToE claim clarified in the correspondence is that constants may be regime quantities rather than eternal primitives.

Thus:

c = c(S,\rho_S,\chi_S,\text{epoch})

where:

• = entropy field level
• = entropy density
• = field responsiveness

Under this interpretation, today’s measured is stable because today’s cosmic entropic phase is stable.

This places ToE conceptually closer to emergent constants programs than to strict immutable constant frameworks.

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8. The Obidi Action as Foundational Necessity

Daniel’s questions repeatedly implied an important challenge:

A theory cannot remain metaphorical forever.

Thus ToE requires an action principle.

The proposed Obidi Action serves this role:

\mathcal{S}_O = \int d^4x \sqrt{-g}\left[
\frac{\alpha}{2}(\partial S)^2 - V(S) + \beta \mathcal{R}_{\text{ent}}(S) + \mathcal{L}_m^{\text{eff}}
\right]

Interpretation:

• kinetic term for entropy field dynamics
• potential term selecting phases
• emergent curvature coupling
• matter as effective excitations

The correspondence reveals this was not decorative mathematics—it was demanded by conceptual pressure.

---

9. The Vuli-Ndlela Integral and History Selection

Another recurring foundational component is the Vuli-Ndlela Integral, conceived as an entropy-constrained generalization of path summation.

Schematically:

Z = \int \mathcal{D}\phi \;
e^{iS[\phi]/\hbar}
e^{-\Sigma[\phi]}

where penalizes entropy-inadmissible histories.

Thus the universe does not merely explore all histories equally; it weights them by irreversible feasibility.

Applied cosmologically:

• histories producing coherent structure dominate,
• runaway inconsistent histories are suppressed,
• expansion trajectories become selected paths.

Daniel’s cosmological questions therefore touched a central pillar of ToE.

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10. Philosophical Depth of the Dialogues

These communications implicitly wrestled with three ancient metaphysical questions.

10.1 What is Space?

Not container, but relation.

10.2 What is Time?

Not parameter, but ordered irreversibility.

10.3 What is Law?

Not imposed command, but stable entropic regularity.

This moves physics from substance ontology toward process ontology.

---

11. Comparison with Historical Transitions

Newton

Space and time absolute.

Einstein

Geometry dynamical.

Quantum Theory

Measurement probabilistic.

ToE Proposal

Entropy prior to geometry, causality, and probability.

Whether correct or not, that is a genuinely foundational move.

---

12. Critical Scientific Challenges Exposed by the Correspondence

The exchanges also illuminate what ToE must still solve.

12.1 Recover Lorentz Symmetry

Show mathematically why emergent entropic dynamics mimic exact Lorentz invariance.

12.2 Derive Einstein Gravity

Obtain GR as a low-energy effective limit.

12.3 Define Microscopic Degrees of Freedom

What physically carries the entropy field?

12.4 Produce Unique Predictions

Without this, ToE remains interpretive rather than predictive.

12.5 Explain Quantum Statistics

How probabilities emerge from entropy geometry.

Daniel’s probing style indirectly highlighted these necessities.

---

13. Sociological Importance of Daniel Alemoh’s Role

Many speculative theories fail because supporters offer only praise.

Daniel’s value lay elsewhere:

• identifying pressure points,
• forcing distinctions,
• asking physically literate questions,
• preserving cordial rigor.

Such correspondents are rare and historically important.

---

14. Deep Assessment of the ToE Program Through These Dialogues

From the reconstructed communications, ToE appears strongest when:

• reinterpreting known principles conceptually,
• distinguishing local vs global dynamics,
• offering ontology-first alternatives.

It appears weakest where all young theories are weak:

• explicit derivations,
• experimental uniqueness,
• microscopic completion.

That is a fair scholarly assessment.

---

15. Conclusion

The communications between Daniel Moses Alemoh and John Onimisi Obidi represent more than private exchanges. They are the anatomy of a theory under formation.

Daniel’s question about superluminal recession versus entropic light-speed limits was not peripheral. It penetrated the deepest structural issue of any emergent-space theory:

How can local causal bounds coexist with global expansion?

The answer developed in the dialogue—that propagation and manifold evolution are categorically distinct—may be one of the clearest conceptual clarifications produced in the ToE correspondence.

Whether the Theory of Entropicity becomes a lasting scientific framework or remains an ambitious speculative program, these dialogues demonstrate a timeless principle:

Major theories begin not in textbooks, but in difficult conversations.

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Acknowledgment

The author acknowledges the vibrant communications of Daniel Moses Alemoh (danielalemoh2@gmail.com) with profound indebtedness and gratitude, especially for his thoughtful and intellectually serious engagement with the developing Theory of Entropicity (ToE), and for posing questions that sharpened its foundational articulation.

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Author Note

John Onimisi Obidi  (jonimisiobidi@gmail.com) is the originator of the Theory of Entropicity (ToE), an entropy-first framework seeking to reformulate the conceptual foundations of modern theoretical physics.

Communications Between Daniel Moses Alemoh and John Onimisi Obidi on the Foundations and Formulation of the Theory of Entropicity (ToE): Dialogues on a New Theory of the Foundation of Modern Theoretical Physics—Part I

Preamble 

This paper presents a structured reconstruction of intellectual communications between Daniel Moses Alemoh (danielalemoh2@gmail.com) and John Onimisi Obidi (jonimisiobidi@gmail.com) concerning the conceptual foundations, physical meaning, and mathematical ambitions of the Theory of Entropicity (ToE). These dialogues centered on a radical proposition: that entropy is not merely a thermodynamic statistic, but a fundamental ontological field from which spacetime structure, causality, measurement, motion, and physical law emerge. The exchanges explored the reinterpretation of the speed of light as an entropic redistribution limit, the meaning of cosmic expansion in an entropy-first cosmology, the status of relativity under an emergent framework, and the role of the Obidi Action and Vuli-Ndlela Integral (VNI) in establishing a new foundational formalism. The correspondence illustrates how rigorous private dialogue can serve as an incubator for theoretical innovation. Beyond historical record, the present work offers a coherent exposition of the evolving logic of the Theory of Entropicity (ToE) and its possible significance for modern theoretical physics.

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1. Introduction

Throughout the history of science, transformative ideas have often matured through correspondence: Newton and Hooke, Einstein and Besso, Bohr and Einstein, Schrödinger and Planck. Informal yet serious intellectual exchanges frequently clarify, sharpen, and test ideas before formal publication.

The present paper documents and synthesizes communications between Daniel Moses Alemoh and John Onimisi Obidi regarding the Theory of Entropicity (ToE), an emerging framework proposing that entropy constitutes the most primitive physical field of reality.

The central reversal proposed by ToE is concise:

Standard physics: geometry, matter, and dynamics are primary; entropy is derivative.
ToE: entropy is primary; geometry, matter, and dynamics are emergent.

Daniel Alemoh’s correspondence was especially significant because it did not merely praise the theory—it probed internal consistency, cosmological implications, and the meaning of physical constants within the framework.

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2. Historical Context of the Correspondence

The exchanges took place during the developmental phase of ToE, when several key constructs had already been proposed:

• The Entropic Field Axiom
• The Obidi Action
• The Master Entropic Equation (MEE)
• The Obidi Field Equations (OFE)
• The Vuli-Ndlela Integral
• The No-Rush Theorem
• The No-Go Theorem
• The Obidi Curvature Invariant (OCI)
• Entropic reinterpretations and reconstructions of relativity, gravitation, and quantum measurement

Daniel Moses Alemoh engaged these ideas critically, especially the claim that the speed of light may be emergent from the dynamical limits of the entropic field.

---

3. The Central Question Raised by Daniel Alemoh

One of the most consequential communications concerned the status of the speed of light .

Daniel correctly interpreted ToE as proposing that:

• is the maximum rate at which the entropic field can reorganize information or energy,
• causal propagation is constrained by the finite response capacity of the field,
• this differs from treating as a brute primitive constant.

He then posed a deeper cosmological challenge:

If space itself emerges from the entropic field, how should cosmic expansion be understood—especially cases where distant galaxies recede effectively faster than ?

This question was profound because it targeted a critical junction between:

• local causality,
• emergent geometry,
• cosmological expansion,
• and the interpretation of constants.

---

4. Daniel Alemoh’s Proposed Resolution

Daniel suggested an elegant distinction:

• The speed of light governs internal reconfiguration within the field.
• Cosmic expansion may instead represent growth or extension of the field itself.

He expressed this metaphorically:

• Light is the fastest ripple through the field.
• Expansion is the field itself increasing its extent.

This insight parallels, in ToE language, the distinction between:

• propagation on a manifold, and
• evolution of the manifold itself.

This was an important conceptual advance because it prevented confusion between:

1. Motion through emergent space
2. Evolution of emergent space

---

5. Obidi’s Reply Within the ToE Framework

In response, the position clarified was that ToE naturally distinguishes two sectors:

5.1 Local Dynamical Sector

This governs:

• particles,
• signals,
• forces,
• measurement,
• causal influence.

Within this domain, the entropic speed limit applies.

5.2 Global Background Sector

This governs:

• entropy vacuum evolution,
• manifold restructuring,
• cosmic expansion,
• large-scale entropy production,
• changing relational geometry.

Thus superluminal recession need not violate causality. It can be interpreted as the entropic manifold re-scaling rather than matter outrunning local entropic transfer limits.

---

6. Reinterpreting the Speed of Light

A recurring clarification in the correspondence was that ToE does not treat as eternally fixed in principle.

Rather:

• is the presently realized maximum redistribution speed of the entropic field.
• Its observed constancy reflects the current regime of the field.
• If the field’s dynamical capacity changed, the effective limiting speed could differ.

This is a strong departure from orthodox relativity, where is fundamental and invariant.

In ToE:

Relativity becomes an emergent regime of a deeper entropic substrate.

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7. The Obidi Action and the Need for Foundations

Another recurring theme was the need to formalize ToE mathematically.

The proposed Obidi Action was conceived as an entropy-first analog of the Einstein–Hilbert action. Symbolically:

\mathcal{S}_{O} = \int d^4x \sqrt{-g}\,\mathcal{L}(S,\partial S,\Phi,\Psi)

where:

• is the entropic field,
• denote emergent matter or coupling sectors.

Its purpose is to derive:

• field equations,
• entropic geodesics,
• effective geometry,
• dynamical constants,
• and cosmological evolution.

Daniel’s questions repeatedly highlighted the necessity of separating intuitive philosophy from operational mathematics.

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8. The Vuli-Ndlela Integral as Cosmological Selector

The correspondence also touched the role of the Vuli-Ndlela Integral, which in ToE modifies path integral reasoning by entropy admissibility and irreversibility weighting.

Conceptually:

• not all histories are equally realized,
• entropy-compatible histories dominate,
• cosmic evolution may follow paths of maximal distinguishability under finite constraints.

Hence expansion itself may be the preferred large-scale entropic history.

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9. Why These Dialogues Matter

Scientific theories rarely emerge fully formed. They are sharpened through criticism.

Daniel Moses Alemoh’s role in these communications was valuable because he repeatedly asked questions at structurally important points:

• What exactly is in ToE?
• How can expansion exceed ?
• Is emergent space compatible with local causality?
• Are equations consistent with interpretation?

Such questions forced greater precision.

This is the hallmark of productive scientific dialogue.

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10. Philosophical Significance

The exchanges reveal that ToE is not merely another modified gravity proposal. It is an attempt to reorder metaphysics:

Instead of:

• objects in space,
• evolving through laws,

ToE suggests:

• entropy-field distinctions generate lawful structure,
• geometry is secondary,
• time reflects irreversible constraint,
• matter is stabilized entropic organization.

This moves physics toward an ontology of process rather than substance.

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11. Challenges Ahead

The correspondence also implicitly reveals unresolved tasks:

Mathematical Tasks

• Derive Lorentz symmetry from entropic principles
• Derive Einstein equations as effective limits
• Predict measurable deviations
• Formalize variable- regimes consistently

Empirical Tasks

• Cosmological signatures
• Timing anomalies
• Quantum measurement delays
• Entropic lensing or propagation effects

Conceptual Tasks

• Define entropy independent of coarse-graining
• Specify microscopic degrees of freedom
• Connect thermodynamic and geometric entropy

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12. Conclusion

The communications between Daniel Moses Alemoh and John Onimisi Obidi represent more than private exchanges. They model how new theories are tested in their formative stages: by curiosity, skepticism, and constructive challenge.

Daniel’s questions concerning the speed of light and cosmic expansion exposed a central tension that helped refine the Theory of Entropicity. Obidi’s responses clarified a crucial distinction between local propagation limits and global manifold evolution.

Whether ToE ultimately succeeds or fails as a physical theory, these dialogues demonstrate an enduring truth of science:

New foundations are built first in conversation.

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Acknowledgment

The author acknowledges Daniel Moses Alemoh (danielalemoh2@gmail.com) with delightful thanks and gratitude for thoughtful engagement, serious criticism, and intellectually honest dialogue during the formative development of the Theory of Entropicity (ToE).

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Author Note

John Onimisi Obidi is the originator of the Theory of Entropicity (ToE), an entropy-first framework seeking to reinterpret and reconstruct the foundations of physics.