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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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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

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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.

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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.

The Haller-Obidi Structural Reduction Theorem of the Theory of Entropicity (ToE)

Theorem 5.1 (Haller-Obidi Structural Theorem). (Structural Reduction).

The Obidi Action of the Theory of Entropicity (ToE) reduces to the Haller-Obidi Action upon localization of the entropic field to a single timelike worldline.

This expresses the formal reduction — the mathematical statement that the Obidi Action reduces to the Haller–Obidi Action under worldline localization. It’s a structural theorem, not a correspondence. It provides the explicit reduction chain:

S_Obidi→S_ent→S_action,

and the conversion H=(2/ℏ)S_HO.

Note:

Theorem 5.1 formalizes the reduction underlying Proposition 5.1, establishing the structural basis of the Obidi–Haller Correspondence.

References 

1)

https://app.clickup.com/9011914914/docs/8cjdu52-591/8cjdu52-611

2)

https://app.clickup.com/9011914914/docs/8cjdu52-591/8cjdu52-611

3)

https://app.clickup.com/9011914914/docs/8cjdu52-591

4)

GitHub/Cloudflare: 

1) https://entropicity.github.io/Theory-of-Entropicity-ToE/papers/

2) https://entropicity.github.io/Theory-of-Entropicity-ToE/docs/The-Theory-of-Entropicity-(ToE)-Living-Review-Letters-Series-Letter-IB-On-the-Haller-Obidi-Action-and-Lagrangian-U1.pdf

Zenodo: 

1) https://doi.org/10.5281/zenodo.19660059

OSF:

1) https://osf.io/5p74x/wiki?wiki=

ORCID Identifier: https://orcid.org/0009-0004-3606-3182

THE THEORY OF ENTROPICITY (ToE) - LIVING REVIEW LETTERS SERIES, Letter IB: On the Haller-Obidi Action and Lagrangian: An Examination of the Mathematical and Conceptual Connection Between John Haller's Action-as-Entropy Equivalence and the Entropic Field Obidi Action Formulation of the Theory of Entropicity (ToE)

GitHub/Cloudflare: 

1) https://entropicity.github.io/Theory-of-Entropicity-ToE/papers/

2) https://entropicity.github.io/Theory-of-Entropicity-ToE/docs/The-Theory-of-Entropicity-(ToE)-Living-Review-Letters-Series-Letter-IB-On-the-Haller-Obidi-Action-and-Lagrangian-U1.pdf

Zenodo: 

1) https://doi.org/10.5281/zenodo.19660059

OSF:

1) https://osf.io/5p74x/wiki?wiki=

ORCID Identifier: https://orcid.org/0009-0004-3606-3182

John Onimisi Obidi

jonimisiobidi@gmail.com

Research Lab, The Aether

April 20, 2026

Category: Research Letter — Theoretical Physics, Information Geometry, Information Theory & Entropic Dynamics

“The principle of least action is the most general and the most powerful method known for the formulation of the laws of physics.” 

— Richard P. Feynman, The Feynman Lectures on Physics (1964)

“The laws of physics must be such that they apply to a world in which information is the fundamental currency.” 

— John Archibald Wheeler, It from Bit (1989)

“Entropy is a measure of our ignorance of the microscopic state of the system.” 

— Edwin T. Jaynes, Information Theory and Statistical Mechanics (1957)

“The gravitational field equations can be viewed as an equation of state, arising from the thermodynamics of spacetime.”

 — Ted Jacobson, Thermodynamics of Spacetime (1995)

Keywords: Theory of Entropicity (ToE); Haller-Obidi Action; Haller-Obidi Lagrangian; Obidi-Haller Correspondence; Entropy-Action Equivalence; Entropic Lagrangian; Covariant Entropic Mechanics; Entropic Field Theory; Mutual Information Geometry; Vuli-Ndlela Integral; Entropic Path Integral; Information Geometry; Fisher-Rao Metric; α-Connection; Quantum Diffusion; Bernoulli Process; Hirshman Entropy; Gaussian Channel; Conditional Entropy; Self-Information; Entropic Dynamics; Least Action; Maximum Entropy; Entropic Flux; OPCEF; Emergent Spacetime

Publication Citation:

Obidi, John Onimisi. (April 20, 2026). ToE Living Review Letters IB: On the Haller-Obidi Action and Lagrangian — An Examination of the Mathematical and Conceptual Connection Between John Haller's Action-as-Entropy Equivalence and the Entropic Field Obidi Action Formulation of the Theory of Entropicity (ToE). Theory of Entropicity (ToE) — Living Review Letters Series. Letter IB.

ABSTRACT

This Letter [Letter IB in the Theory of Entropicity (ToE) Living Review Letters Series] presents a rigorous mathematical examination of the structural and formal connections between John L. Haller Jr.'s 2015 entropy-action identity — H = (2/ℏ)∫(mc² − L)dt — and the Obidi entropic field action formulation of the Theory of Entropicity (ToE). We define the Haller-Obidi Action as the explicit single-particle entropic action SHO = ∫ℒHO dt whose Lagrangian ℒHO = mc² − (ℏ/2)(dH/dt) is constructed by rearranging Haller's central result into a variational form that ToE absorbs as a worldline sector. We demonstrate that this Haller-Obidi Lagrangian admits a natural covariant generalization ℒent = mc² − (ℏ/2)(uμ ∂μ S) when the entropic field S(x) of ToE is restricted to a particle worldline. The formal reduction of the Obidi Action to the Haller-Obidi Action is established through a localization procedure, proving that Haller's identity is the single-particle projection of the universal entropic field dynamics. We further show that Haller's decomposition H = HC + IM maps onto the free-plus-interaction decomposition of the entropic Lagrangian, that the mutual information rate dIM/dt = (2/ℏ)V provides a concrete prototype for entropic coupling constants and information-geometric potentials, and that the Gaussian channel structure underlying Haller's derivation corresponds to the α = 0 (Levi-Civita) sector of ToE's entropic α-connection. We explore the bridge to the Vuli-Ndlela Integral through entropy-weighted path selection, and we identify the precise mathematical limits of the Haller-ToE correspondence — including the absence of an entropic field, conserved entropic flux, and intrinsic time asymmetry in Haller's framework. The Haller-Obidi Action and Lagrangian thus serve as a concrete, calculable bridge between information-theoretic particle mechanics and the full entropic field theory of ToE.

++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++

This Letter [Letter IB in the Theory of Entropicity (ToE) Living Review Letters Series] presents a rigorous mathematical examination of the structural and formal connections between John L. Haller Jr.'s 2015 entropy-action identity — H = (2/ℏ)∫(mc² − L)dt — and the Obidi entropic field action formulation of the Theory of Entropicity (ToE). We define the Haller-Obidi Action as the explicit single-particle entropic action SHO = ∫ℒHO dt whose Lagrangian ℒHO = mc² − (ℏ/2)(dH/dt) is constructed by rearranging Haller's central result into a variational form that ToE absorbs as a worldline sector. We demonstrate that this Haller-Obidi Lagrangian admits a natural covariant generalization ℒent = mc² − (ℏ/2)(uμ ∂μ S) when the entropic field S(x) of ToE is restricted to a particle worldline. The formal reduction of the Obidi Action to the Haller-Obidi Action is established through a localization procedure, proving that Haller's identity is the single-particle projection of the universal entropic field dynamics. We further show that Haller's decomposition H = HC + IM maps onto the free-plus-interaction decomposition of the entropic Lagrangian, that the mutual information rate dIM/dt = (2/ℏ)V provides a concrete prototype for entropic coupling constants and information-geometric potentials, and that the Gaussian channel structure underlying Haller's derivation corresponds to the α = 0 (Levi-Civita) sector of ToE's entropic α-connection. We explore the bridge to the Vuli-Ndlela Integral through entropy-weighted path selection, and we identify the precise mathematical limits of the Haller-ToE correspondence — including the absence of an entropic field, conserved entropic flux, and intrinsic time asymmetry in Haller's framework. The Haller-Obidi Action and Lagrangian thus serve as a concrete, calculable bridge between information-theoretic particle mechanics and the full entropic field theory of ToE.

EXECUTIVE SUMMARY

●      Haller (2015) derives H = (2/ℏ)∫(mc² − L)dt from first principles in information theory and quantum diffusion, directly identifying entropy with the classical action. The derivation proceeds through a Bernoulli-Gaussian diffusion model, the Hirshman entropy sum, conditional entropy rates, and a Gaussian mutual information channel — each contributing one structural element to the final identity.

●      We construct the Haller-Obidi Lagrangian ℒHO ≡ mc² − (ℏ/2)Ḣ by rearranging Haller's central result, yielding an explicit entropic effective action at the particle level that admits variational treatment. The resulting Haller-Obidi Action SHO = ∫ℒHO dt is shown to be identically equal to the classical action Saction, confirming internal consistency while exposing the informational anatomy of the classical Lagrangian.

●      The covariant generalization ℒent = mc² − (ℏ/2)(uμ ∂μ S) connects the Haller-Obidi construction to the entropic field S(x) of ToE, with the entropic current JμS = ρS uμ and the continuity condition ∇μ JμS = 0 emerging naturally. The covariant formulation extends Haller's non-relativistic identity to arbitrary curved spacetimes.

●      The Obidi Action SObidi = ∫F(S, ∇S, gμν) d⁴x reduces to the Haller-Obidi Action upon worldline localization, establishing the Obidi-Haller Correspondence as a rigorous mathematical limit: Obidi Action (field level) → Haller-Obidi Action (worldline level) → Classical Action (non-relativistic limit).

●      Haller's mutual information rate dIM/dt = (2/ℏ)V provides a prototype for information-geometric potentials and suggests a route from mutual information to an effective entropic metric g(ent)μν ~ ∂²IM/∂θμ∂θν, realizing the Fisher-Rao metric as emergent spacetime geometry in the α = 0 sector of ToE's entropic α-connection.

●      The entropy-weighted path selection implicit in Haller's framework motivates the Vuli-Ndlela path integral ZVN = ∫𝒟[x] exp{iS[x]/ℏ + λH[x]}, while the limits of the correspondence are clearly delineated: Haller does not construct an entropic field, conserved flux, or intrinsic time asymmetry — structures that emerge only at the full field-theoretic level of the Obidi Action.

1.   Introduction — From Entropy-Action Identity to Entropic Lagrangian Mechanics

The principle of least action and the concept of entropy have been treated as conceptually distinct pillars of physics for over three centuries. The action functional — Hamilton's integral of the Lagrangian along a worldline — governs the trajectories of particles and the dynamics of fields through a variational principle that selects, from the space of all kinematically admissible histories, the unique path that extremizes the action. Entropy, by contrast, enters physics through the second law of thermodynamics and its information-theoretic generalizations: it measures the multiplicity of microstates consistent with a given macrostate, the uncertainty in a probability distribution, the irreversibility of a dynamical process. Action selects trajectories; entropy counts states. Action is reversible; entropy is directional. Action lives in configuration space; entropy lives in probability space. Or so the canonical wisdom has maintained.

This canonical separation was first challenged at the interface of general relativity and quantum field theory. Bekenstein's 1973 derivation of black hole entropy [2], proportional to the horizon area in Planck units, demonstrated that gravitational dynamics encodes information-theoretic content in its geometry. Jacobson's 1995 thermodynamic derivation of the Einstein field equations from the Clausius relation δQ = T δS applied to local Rindler horizons [3] showed that spacetime curvature could be understood as a macroscopic consequence of microscopic entropic dynamics. Verlinde's 2011 entropic gravity program [4] and Padmanabhan's surface-bulk thermodynamic framework [5] extended this insight, arguing that gravitational acceleration itself is an entropic force emerging from the statistical mechanics of horizon degrees of freedom. Frieden's Fisher information approach [6] and Jaynes' maximum entropy formalism [7] attacked the problem from the information-theoretic side, showing that the equations of motion of classical and quantum mechanics could be derived from optimization principles on probability distributions.

Letter IA of this series — The Entropic Rosetta Stone [15] — surveyed this historical and conceptual landscape in detail, establishing the position of the Theory of Entropicity (ToE) as the synthesis and extension of these entropy-as-generator programs. The central claim of Letter IA was that the tradition of deriving dynamics from entropy is not merely a collection of independent results but the partial excavation of a single underlying structure: the entropic field S(x) of ToE, whose dynamics generate geometry, fields, and law from a single informational primitive.

Within that survey, the 2015 paper by John L. Haller Jr., "Action as Entropy" [1], occupied a position of special importance. Haller demonstrated, through a self-contained derivation grounded in quantum diffusion, Bernoulli processes, and information-theoretic entropies, that the total self-information of a quantum particle — defined as the sum of conditional entropy and mutual information — equals the time integral of the mass-energy minus the classical Lagrangian, scaled by the quantum of action:

H = (2/ℏ) ∫ (mc² − L) dt…………………………………………………………………………………………………. (1)

This is the entropy-action identity: a direct mathematical equation between an information-theoretic quantity (the Hirshman entropy of a quantum diffusion process) and a mechanical quantity (the classical action plus a rest-energy baseline). Letter IA discussed the conceptual significance of this result for ToE. The present Letter — Letter IB — has a different and more specific objective.

The objective of this Letter is to examine the precise mathematical and structural connections between Haller's particle-level entropy-action identity and the entropic field action formulation of the Theory of Entropicity. Where Letter IA asked, "What does Haller's result mean for ToE?", Letter IB asks: "What mathematical structures connect Haller's particle-level identity to the Obidi field-level action, and what new constructions emerge from their synthesis?"

The answer, as we shall demonstrate, is rich. From Haller's identity (1) and the Obidi Action of ToE:

SObidi = ∫ F(S, ∇S, gμν) √(−g) d⁴x………………………………..………………………………………………….. (2)

we construct two named mathematical objects that serve as the connective tissue between the particle-level and field-level formulations:

(i) The Haller-Obidi Lagrangian, ℒHO ≡ mc² − (ℏ/2)Ḣ, obtained by rearranging Haller's entropy rate identity into a variational object — a Lagrangian in the mechanical sense that can be subjected to the Euler-Lagrange procedure, yielding equations of motion that are simultaneously mechanical and informational.

(ii) The Haller-Obidi Action, SHO ≡ ∫ℒHO dt, the time integral of the Haller-Obidi Lagrangian, which we show is identically equal to the classical action. The classical action is the entropic action, rewritten in information-theoretic variables.

Beyond these definitions, we construct the covariant generalization of the Haller-Obidi Lagrangian by embedding the non-relativistic entropy rate into the entropic field S(x) of ToE, yielding a worldline Lagrangian that couples particle motion to the ambient entropic field through the four-velocity contraction uμ ∂μ S. We prove that the Obidi Action (2) reduces to this covariant Haller-Obidi Action upon localization of the entropic field to a single timelike worldline — establishing the Obidi-Haller Correspondence as a rigorous mathematical limit, not merely an analogy.

We further demonstrate that Haller's information-theoretic decomposition H = HC + IM maps precisely onto the free-plus-interaction decomposition of the entropic Lagrangian; that the mutual information rate dIM/dt = (2/ℏ)V provides a concrete prototype for entropic coupling constants; that the Gaussian channel structure of Haller's mutual information calculation corresponds to the α = 0 (Levi-Civita) sector of ToE's entropic α-connection; and that the entropy-weighted path selection implicit in Haller's framework motivates the Vuli-Ndlela path integral of ToE. We close by stating honestly and precisely where the mathematical correspondence ends — what structures of ToE have no counterpart in Haller's framework, and what structures of Haller's framework do not survive the passage to the full entropic field theory.

The central task of this Letter is therefore to construct the precise mathematical bridge between equations (1) and (2). The construction will proceed in stages: we first reconstruct the mathematical anatomy of Haller's derivation (Section 2), then define and analyze the Haller-Obidi Lagrangian and Action (Section 3), construct the covariant generalization (Section 4), prove the reduction from the Obidi Action (Section 5), explore the information-geometric and path-integral bridges (Sections 6–8), and delineate the limits of the correspondence (Section 9).

10.   Conclusion

This Letter (Letter IB in the Theory of Entropicity (ToE) Living Review Letters Series) has established the precise mathematical and structural connections between John L. Haller Jr.'s 2015 entropy-action identity and the entropic field action formulation of the Theory of Entropicity (ToE). The analysis has introduced several new mathematical constructions, each serving a specific function in the bridge between particle-level information theory and field-level entropic dynamics:

1. The Haller-Obidi Lagrangian ℒHO = mc² − (ℏ/2)Ḣ (Definition 3.1, equation (13)) — the explicit entropic Lagrangian at the single-particle level, obtained by rearranging Haller's entropy-rate identity into a variational form. This Lagrangian admits Euler-Lagrange treatment and encodes the dual mechanical-informational character of classical trajectories.

2. The Haller-Obidi Action SHO = ∫ℒHO dt = Saction (Definition 3.2, equations (14)–(17)) — the time integral of the Haller-Obidi Lagrangian, shown to be identically equal to the classical action. The classical action is the entropic action, expressed in information-theoretic variables. This identity is exact within Haller's non-relativistic framework.

3. The covariant Haller-Obidi Lagrangian ℒent = mc² − (ℏ/2)(uμ ∂μ S) (Definition 4.1, equation (24)) — the generally covariant extension of the Haller-Obidi Lagrangian, coupling particle motion (via four-velocity uμ) to the ambient entropic field (via ∂μ S). This Lagrangian lives on arbitrary pseudo-Riemannian manifolds and reduces to ℒHO in the non-relativistic limit.

4. The Obidi-Haller Correspondence (Proposition 5.1, equations (38)–(40)) — the formal demonstration that the Obidi Action reduces to the covariant Haller-Obidi Action upon worldline localization of the entropic field. This establishes the hierarchy: Obidi Action (field level) → Haller-Obidi Action (worldline level) → Classical Action (non-relativistic limit), with each level emerging from the one above by mathematical restriction.

5. The information-geometric bridge (Section 6, equation (44)) — the identification of Haller's mutual information rate with an effective Fisher-Rao metric on the space of vacuum configurations, providing a constructive route from mutual information to the emergent physical metric of ToE. The entropic coupling constants gent (equation (42)) parameterize the informational strength of fundamental interactions.

6. The Vuli-Ndlela bridge (Section 8, equations (48)–(49)) — the connection between Haller's entropy-weighted path selection and the Vuli-Ndlela path integral of ToE. Haller's result demonstrates that the standard Feynman path integral already contains an implicit entropy weighting; the Vuli-Ndlela Integral makes this explicit and generalizes it through the entropic selection parameter λ, introducing an intrinsic time asymmetry that the standard formulation lacks.

These six constructions demonstrate that Haller's entropy-action identity and the Obidi entropic field theory are not merely analogous or philosophically aligned — they are mathematically nested. The Haller-Obidi Action is the single-particle projection of the Obidi Action. The classical Lagrangian is the informational residue of the entropic field evaluated along a worldline. The information-theoretic decomposition H = HC + IM is the single-particle projection of the field-theoretic decomposition ℒ = ℒfree + ℒint. The mutual information rate is the single-particle value of the Fisher-Rao metric. At every level of description, the particle-level information theory and the field-level entropic dynamics map onto each other through precisely defined mathematical correspondences.

The limits of the correspondence are equally precise. Haller does not construct an entropic field, conserved flux, field equations, or intrinsic time asymmetry — these are ToE structures that emerge only at the field-theoretic level. But within his domain of validity (single particle, non-relativistic, Gaussian channel, α = 0 sector), Haller's results are exact and provide an independently derived confirmation of ToE's central claim: that entropy and action are two faces of the same mathematical structure.

Several directions for future work emerge naturally from this analysis. First, the many-body generalization of the Haller-Obidi Lagrangian — the extension from a single particle to a system of interacting particles, with mutual information between all pairs — would provide the first concrete multi-particle sector of the Obidi Action. Second, the fully relativistic extension of Haller's derivation — replacing the Taylor expansions in v/c with exact Lorentz-covariant expressions — would establish the Haller-Obidi correspondence at all velocities. Third, the explicit derivation of the Haller-Obidi Action from the Obidi Field Equations — constructing the localized solution S(x) that, upon worldline restriction, yields exactly the Haller-Obidi Lagrangian — would elevate the Obidi-Haller Correspondence from an ansatz-dependent result to a theorem of the Obidi Field Equations. Fourth, the experimental signatures of the entropic selection parameter λ — the correlations between irreversibility and transition-amplitude deviations predicted by the Vuli-Ndlela Integral — offer a concrete avenue for empirical tests of the entropic selection principle.

The Haller-Obidi Action and Lagrangian, defined and analyzed in this Letter, serve as the mathematical hinge between information-theoretic particle mechanics and the full entropic field theory of the Theory of Entropicity (ToE). They show that the relation between entropy and action—first recognized as a philosophical alignment and later established as a formal identity by Haller—constitutes a genuine structural theorem with precise mathematical content, well‑defined limits, and generative consequences for the foundations and formulation of the Theory of Entropicity (ToE).

The Obidi-Haller Correspondence and Its Significance in Modern Theoretical Physics 

The Obidi–Haller Correspondence is a theoretical framework within the Theory of Entropicity (ToE) that explores the relationship between physical entropy and action. It specifically maps how John Haller's concept of "action-as-entropy" serves as a precursor to and validation of the broader entropic field theories proposed by Obidi. [1]

Key Components

• Entropy–Action Identity: The correspondence demonstrates that Haller's 2015 identity (where action is equivalent to entropy) emerges as the "single particle limit" of the more complex Obidi entropic field.
• Theoretical Significance: It is often referred to as an "Entropic Rosetta Stone" because it provides a comparative bridge between earlier entropy-action equivalence traditions and modern Theory of Entropicity (ToE).
• Authorship: The framework is associated with the work of researchers like John Haller and the development of the Theory of Entropicity (ToE) Living Review Letters Series. [1]

Would you like to dive deeper into the mathematical formulas used in this correspondence or explore its applications in quantum physics?

 

[1] https://www.academia.edu

 

The Obidi–Haller Correspondence is a concept within the "Theory of Entropicity" (ToE) that explains how John Haller's "entropy–action identity" emerges as the single-particle limit of the broader Obidi entropic field.

Core Concept: It serves as a bridge between Haller's 2015 work, which posits that action is equivalent to entropy, and the more comprehensive Theory of Entropicity.

Significance: It is analyzed within the "Entropic Rosetta Stone" as a key to validating how individual particle actions connect to overall entropic fields.

Context: This correspondence is positioned within a deeper analysis of the entropy-action equivalence tradition. 

References 

GitHub/Cloudflare: 

1) https://entropicity.github.io/Theory-of-Entropicity-ToE/papers/

2) https://entropicity.github.io/Theory-of-Entropicity-ToE/docs/The-Theory-of-Entropicity-(ToE)-Living-Review-Letters-Series-Letter-IB-On-the-Haller-Obidi-Action-and-Lagrangian-U1.pdf

Zenodo: 

1) https://doi.org/10.5281/zenodo.19660059

OSF:

1) https://osf.io/5p74x/wiki?wiki=

ORCID Identifier: https://orcid.org/0009-0004-3606-3182