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)

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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.

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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).