John Haller and John Obidi: The Haller–Obidi Correspondence — How Obidi Reinterprets Classical Physics as Entropic Physics in the Theory of Entropicity (ToE)
Introduction
The Haller–Obidi Correspondence occupies a central conceptual and mathematical position within the developing Theory of Entropicity (ToE). Its significance arises from the proposed correspondence between the classical action functional and an information-theoretic or entropic quantity, and from Obidi's subsequent effort to promote that correspondence from a particle-level relation into a general field-theoretic principle.
The essential idea is that the quantity ordinarily called the classical action need not be regarded as a fundamentally separate dynamical object from entropy or self-information. If the Haller relation
$$
H = \frac{2}{\hbar}\int \left(mc^2-\mathcal{L}\right),dt
$$
is taken as the starting point, then the classical action can be recast in informational terms. Obidi's contribution within ToE is to follow this correspondence to its logical consequences: entropy is promoted from a descriptive quantity associated with thermodynamic states to a dynamical field quantity, and the variational principles of mechanics are consequently reconsidered as manifestations of an underlying principle of extremal entropy.
The resulting program is considerably broader than a simple reinterpretation of an existing equation. Obidi seeks to establish a chain of correspondence
$$
\text{information}
\longleftrightarrow
\text{entropy}
\longleftrightarrow
\text{action}
\longleftrightarrow
\text{geometry}
\longleftrightarrow
\text{dynamics},
$$
in which the familiar structures of physics emerge as different mathematical expressions of a more fundamental entropic geometry.
The Haller–Obidi Correspondence therefore functions as a proposed bridge between two descriptions that are conventionally treated as belonging to different conceptual domains: the variational formulation of mechanics and the informational formulation of physical systems.
It is important to distinguish here between the historical or established status of the mathematical relations involved and their interpretation within ToE. The present treatment describes the Haller relation and the subsequent Obidi construction as components of the ToE research program. The broader claim that all physical dynamics are fundamentally entropic is a theoretical proposition of ToE and therefore requires its own mathematical derivation and, ultimately, empirical validation.
The methodological position underlying this research is equally important. Obidi's objective is not primarily to make the developing theory conform to existing academic consensus. His objective is to determine what follows when the underlying mathematical insight is pursued rigorously and consistently to its logical conclusion. Independence from consensus is therefore not independence from mathematical discipline. On the contrary, the more unconventional the hypothesis, the greater the requirement for mathematical precision, dimensional consistency, covariance where appropriate, conservation laws, limiting behavior, and eventual empirical verification.
The governing principle is therefore:
$$
\text{independence from consensus}
\neq
\text{independence from verification}.
$$
---
1. Haller's Entropy–Action Identity
The starting point for the Haller–Obidi Correspondence is the relationship attributed to John Haller between the classical action of a particle and its self-information or entropy.
For a classical particle described by a Lagrangian $\mathcal{L}$, Haller's relation is written as
$$
H
\frac{2}{\hbar}
\int
\left(
mc^2-\mathcal{L}
\right),dt.
$$
The expression can be understood by introducing the classical action
$$
\mathcal{A}
\int \mathcal{L},dt.
$$
Then
$$
H
\frac{2mc^2}{\hbar}t
\frac{2}{\hbar}\mathcal{A}.
$$
Thus, apart from the rest-energy contribution, the entropy-like quantity $H$ is directly related to the classical action. Equivalently, one may write
$$
\mathcal{A}
mc^2t
\frac{\hbar}{2}H.
$$
This is the fundamental structural observation upon which Obidi's extension is built.
The conventional formulation of classical mechanics treats the action
$$
\mathcal{A}=\int \mathcal{L},dt
$$
as the central object of the variational formulation. The physical trajectory is obtained from
$$
\delta\mathcal{A}=0,
$$
which generates the Euler–Lagrange equations
$$
\frac{d}{dt}
\left(
\frac{\partial\mathcal{L}}{\partial\dot q^i}
\right)
\frac{\partial\mathcal{L}}{\partial q^i}
0.
$$
The Haller relation suggests that the same variational structure may be expressible in terms of an entropy-like quantity. In this interpretation, action and entropy are not independent conceptual categories but different representations of a common underlying quantity.
This is the first decisive conceptual transition in ToE:
$$
\text{classical action}
\longrightarrow
\text{entropic/informational quantity}.
$$
The significance of this correspondence is that the action principle already possesses a variational structure. If the action is representable in terms of entropy, then the extremal principle of mechanics can potentially be reformulated as an extremal principle involving entropy.
The resulting interpretation is profound: the variational character of physical law may not be fundamentally mechanical at all. It may be informational.
---
2. From Particle Action to Entropic Dynamics
The importance of the Haller relation within ToE lies not merely in its ability to rewrite one integral in different notation. Its deeper significance is that it provides a possible mechanism for transforming the variational language of mechanics into the language of information and entropy.
In conventional mechanics,
$$
\mathcal{A}[q]
\int
\mathcal{L}(q,\dot q,t),dt
$$
is a functional over possible trajectories. The physical trajectory is selected through an extremal condition,
$$
\delta\mathcal{A}=0.
$$
If an entropy functional $H[q]$ is related to the action by an affine transformation of the form
$$
H[q]
C_1-C_2\mathcal{A}[q],
$$
where $C_1$ and $C_2$ are constants with respect to the trajectory variation, then
$$
\delta H
-C_2,\delta\mathcal{A}.
$$
Consequently,
$$
\delta\mathcal{A}=0
\quad\Longleftrightarrow\quad
\delta H=0.
$$
This is the mathematical basis for the ToE interpretation of the classical variational principle as an entropic extremum principle.
The point is not that the Euler–Lagrange equations disappear. Rather, they can be interpreted as equations describing the stationary trajectories of an entropic functional.
Thus,
$$
\delta\mathcal{A}=0
\quad\Longleftrightarrow\quad
\delta H=0.
$$
under the conditions required by the Haller correspondence.
This is the mathematical doorway through which classical mechanics can be re-expressed as entropic mechanics.
The distinction between a minimum, a maximum, and a general stationary or extremal value is important here. The classical principle of stationary action does not universally require the action to be a minimum. Therefore, the mathematically precise ToE terminology is principle of extremal entropy or principle of stationary entropy, unless the complete theory establishes additional stability conditions that specifically select a maximum or minimum.
---
3. Obidi's Extension to Field Theory
Haller's relation is formulated at the level of a particle trajectory. Obidi's more ambitious step is to extend the correspondence into a field-theoretic framework.
Instead of treating entropy merely as an integrated quantity associated with a particle's history, ToE introduces an entropic field
$$
S(x),
$$
or, in relativistic notation,
$$
S(x^\mu).
$$
Here $S$ is intended to represent an underlying scalar informational or entropic degree of freedom distributed over spacetime or over a more fundamental geometric manifold.
The gradient
$$
\partial_\mu S
$$
then provides the local variation of the entropic field. A particle with four-velocity $u^\mu$ samples this field along its worldline. The directional derivative
$$
u^\mu\partial_\mu S
$$
therefore measures the rate at which the particle encounters or traverses the entropic field along its trajectory.
Obidi introduces the entropic Lagrangian in the form
$$
\mathcal{L}_{\mathrm{ent}}
mc^2
\frac{\hbar}{2}
\left(
u^\mu\partial_\mu S
\right).
$$
The construction is significant because the entropic quantity is no longer merely an integrated consequence of a mechanical trajectory. The entropy field enters the local dynamical description itself.
The quantity
$$
u^\mu\partial_\mu S
$$
is the covariant directional derivative of $S$ along the particle's worldline,
$$
\frac{dS}{d\tau}
u^\mu\partial_\mu S,
$$
provided
$$
u^\mu=\frac{dx^\mu}{d\tau}.
$$
Consequently, the entropic Lagrangian can equivalently be expressed as
$$
\mathcal{L}_{\mathrm{ent}}
mc^2
\frac{\hbar}{2}
\frac{dS}{d\tau},
$$
subject to the precise choice of parameterization and normalization conventions adopted by the theory.
This step transforms the conceptual role of entropy. Entropy is no longer merely a quantity calculated after the dynamics have occurred. It becomes part of the mathematical structure from which the dynamics themselves are formulated.
This is the transition
$$
\text{entropy as derived quantity}
\longrightarrow
\text{entropy as dynamical field}.
$$
It represents one of the principal conceptual advances claimed by the framework.
---
4. The Obidi–Haller Action
Integrating the entropic Lagrangian produces the corresponding entropic action, which may be designated the Obidi–Haller Action (OHA):
$$
\mathcal{A}_{\mathrm{OH}}
\int
\mathcal{L}_{\mathrm{ent}},d\tau.
$$
Thus,
$$
\mathcal{A}_{\mathrm{OH}}
\int
\left[
mc^2
\frac{\hbar}{2}
u^\mu\partial_\mu S
\right]
d\tau.
$$
Using
$$
u^\mu\partial_\mu S
\frac{dS}{d\tau},
$$
one obtains
$$
\mathcal{A}_{\mathrm{OH}}
mc^2\int d\tau
\frac{\hbar}{2}
\int dS.
$$
For a trajectory extending between two endpoints,
$$
\mathcal{A}_{\mathrm{OH}}
mc^2\Delta\tau
\frac{\hbar}{2}\Delta S.
$$
This expression makes the structural relationship particularly transparent. The action contains an entropic contribution proportional to the change in the entropic field along the particle's trajectory.
The construction therefore provides a field-theoretic realization of the original Haller correspondence. Instead of entropy appearing only after the mechanical action has been evaluated, the entropic quantity is incorporated directly into the Lagrangian structure.
In ToE, this is the transition
$$
\text{entropy as derived quantity}
\longrightarrow
\text{entropy as dynamical field}.
$$
At the same time, the precise physical content of the OHA depends critically upon the full ToE field equations. In particular, if $S$ enters the action only through a total derivative, then its variation may reduce to a boundary contribution and may not generate nontrivial bulk dynamics by itself.
Indeed, because
$$
u^\mu\partial_\mu S
\frac{dS}{d\tau},
$$
the term
$$
\int u^\mu\partial_\mu S,d\tau
$$
becomes
$$
\int dS
\Delta S.
$$
Therefore, if the displayed OHA is the complete action and no additional dependence on $S$, $\partial_\mu S$, or higher derivatives is present, the entropic term is a boundary term. A complete ToE formulation must therefore specify the additional terms, couplings, boundary conditions, or geometric structures through which $S$ acquires independent dynamical content.
This is not a weakness of the research program; it identifies one of the mathematical questions that must be resolved as the theory is developed.
A nontrivial field-theoretic entropic action might, for example, contain additional terms schematically of the form
$$
\mathcal{L}_{S}
-\frac{\kappa}{2}
g^{\mu\nu}
\partial_\mu S
\partial_\nu S
V(S),
$$
or more generally
$$
\mathcal{L}_{\mathrm{ToE}}
\mathcal{L}{\mathrm{ent}}
+
\mathcal{L}{S}
+
\mathcal{L}{\mathrm{matter}}
+
\mathcal{L}{\mathrm{geom}}
+
\mathcal{L}_{\mathrm{int}},
$$
although the precise form of these terms must be derived from the actual ToE framework rather than assumed in advance.
This distinction between the worldline entropic coupling and the independent field dynamics of $S$ is essential for developing the correspondence into a genuine field theory.
---
5. The Haller–Obidi Correspondence
The correspondence can now be stated in its strongest form within ToE:
$$
\text{classical action}
\longleftrightarrow
\text{entropic/informational functional}.
$$
and, consequently,
$$
\delta\mathcal{A}=0
\quad\longleftrightarrow\quad
\delta H=0.
$$
The conventional principle of stationary action therefore acquires an informational interpretation.
The usual mechanical statement is
$$
\delta
\int
\mathcal{L},dt
0.
$$
The ToE statement is that the same physical trajectory may be characterized as an extremum of the corresponding entropic functional,
$$
\delta H=0.
$$
Accordingly, the classical trajectory is not regarded merely as the path that makes an action stationary. Within the ToE interpretation, it is the path selected by the extremization of an underlying informational or entropic quantity.
The word extremal is important. The principle should not automatically be described as "maximum entropy" or "minimum entropy." The classical stationary-action principle does not generally specify a universal minimum; it specifies a stationary value. Therefore, the mathematically precise ToE formulation is a principle of extremal entropy or stationary entropy, unless the detailed theory establishes the sign and stability conditions necessary to identify a maximum or minimum.
This distinction becomes increasingly important as ToE is extended into relativistic field theory, quantum theory, and gravitational geometry.
---
6. From Least Action to Extremal Entropy
The conceptual transformation can therefore be expressed as
$$
\text{Principle of Stationary Action}
\quad\longrightarrow\quad
\text{Principle of Extremal Entropy}.
$$
provided that the action and entropy functionals are related by the appropriate transformation.
In conventional mechanics, the action is the organizing functional of dynamics. Its variation produces the equations of motion. ToE proposes that the action may instead be understood as a representation of an underlying entropic structure.
This changes the ontological interpretation of the variational principle.
Rather than saying simply that nature "chooses" a trajectory of stationary action, ToE asks whether the deeper statement is that physical evolution selects configurations that are stationary with respect to an underlying informational or entropic functional.
The distinction is subtle but fundamental.
The standard formulation is
$$
\delta\mathcal{A}=0.
$$
The proposed entropic formulation is
$$
\delta\mathcal{S}_{\mathrm{ent}}=0.
$$
The mathematical challenge for ToE is then to construct
$$
\mathcal{S}_{\mathrm{ent}}
$$
such that the known equations of physics emerge as appropriate limits of its variation.
A successful theory would therefore need to demonstrate not merely a formal similarity between action and entropy, but a systematic derivation of physical dynamics from the entropic functional.
---
7. The Physical Meaning of the Entropic Field $S(x)$
The introduction of
$$
S(x)
$$
is consequential because it changes the status of entropy from a quantity associated with a system to a field capable of possessing local structure.
The gradient
$$
\partial_\mu S
$$
describes the local variation of the entropic field. Its contraction with the four-velocity,
$$
u^\mu\partial_\mu S,
$$
describes the rate at which the entropic state changes along a physical trajectory.
Within the ToE interpretation, this provides a possible mechanism through which information geometry can influence dynamics.
A particle does not simply move through a passive spacetime background. It moves through an informationally structured environment described by $S(x)$. The local geometry and gradient structure of this field can consequently become relevant to the particle's dynamical evolution.
This creates the conceptual sequence
$$
S(x)
\rightarrow
\partial_\mu S
\rightarrow
u^\mu\partial_\mu S
\rightarrow
\mathcal{L}{\mathrm{ent}}
\rightarrow
\mathcal{A}{\mathrm{OH}}
\rightarrow
\text{equations of motion}.
$$
The significance of this sequence is that it potentially provides a route from a scalar informational quantity to observable dynamics.
The deeper ToE question is whether this scalar field is itself fundamental or whether it is an effective representation of a more general informational object. If the theory ultimately associates $S$ with a metric, connection, probability distribution, state-space structure, or higher-dimensional geometric field, then $S(x)$ may be interpreted as one coordinate representation of a more fundamental informational geometry.
---
8. Classical Physics as a Limit of Entropic Physics
One of the larger claims of ToE is that classical mechanics should not necessarily be regarded as a fundamental layer of physics. Instead, it may constitute a limiting representation of a deeper entropic dynamics.
Under this interpretation, Newtonian mechanics, relativistic mechanics, and potentially quantum dynamics are not independent foundational theories. They are different mathematical regimes in which the underlying informational geometry manifests itself in different forms.
The proposed hierarchy is therefore
$$
\text{Entropic foundation}
\rightarrow
\text{informational geometry}
\rightarrow
\text{physical geometry}
\rightarrow
\text{dynamics}.
$$
with classical mechanics appearing as one effective sector of the resulting structure.
This does not imply that Newton's laws cease to be valid within their domain. Rather, it changes their interpretation. A successful ToE would explain why the familiar classical equations emerge from the more fundamental entropic description.
For example, if an entropic action
$$
\mathcal{A}_{\mathrm{ent}}
$$
can be shown to reduce in an appropriate limit to
$$
\mathcal{A}_{\mathrm{cl}}
\int\mathcal{L}_{\mathrm{cl}},dt,
$$
then the classical equations of motion would appear as emergent equations of the entropic theory.
The objective is consequently not to replace classical mechanics arbitrarily, but to identify the deeper structure from which classical mechanics can be recovered.
The same logic applies to relativistic and quantum regimes. The ToE program is strongest when the limiting process can be made explicit:
$$
\mathcal{T}{\mathrm{ToE}}
\xrightarrow[\text{appropriate limit}]{}
\mathcal{T}{\mathrm{classical}},
$$
and, where applicable,
$$
\mathcal{T}{\mathrm{ToE}}
\xrightarrow[\text{appropriate limit}]{}
\mathcal{T}{\mathrm{relativistic}},
$$
or
$$
\mathcal{T}{\mathrm{ToE}}
\xrightarrow[\text{appropriate limit}]{}
\mathcal{T}{\mathrm{quantum}}.
$$
These limits would need to be derived rather than asserted.
---
9. Action, Entropy, Information, and Geometry
The philosophical depth of the Haller–Obidi Correspondence becomes apparent when its implications are followed beyond the immediate action–entropy relation.
Classical physics traditionally assigns different mathematical roles to several concepts. The action is a variational functional. Entropy is associated primarily with statistical and thermodynamic descriptions. Information measures distinguishability, uncertainty, or knowledge of states. Geometry describes the mathematical structure of configuration space, spacetime, or state space.
ToE proposes that these categories may not be fundamentally independent.
The proposed chain is
$$
\text{information}
\rightarrow
\text{entropy}
\rightarrow
\text{geometry}
\rightarrow
\text{action}
\rightarrow
\text{dynamics}.
$$
Alternatively, because the proposed correspondence is bidirectional at the structural level,
$$
\text{action}
\longleftrightarrow
\text{entropy}
\longleftrightarrow
\text{information}
\longleftrightarrow
\text{geometry}.
$$
The important claim is therefore ontological as well as mathematical: physical quantities that are ordinarily introduced as separate primitives may instead represent different projections of a deeper informational structure.
Within this interpretation, geometry is not merely the stage on which physics occurs. Geometry itself may be generated by informational relationships.
Similarly, action is not merely an abstract integral used to derive equations of motion. It may be an integrated representation of the entropic structure governing physical evolution.
Entropy, in turn, is not merely a thermodynamic accounting quantity. It may represent a more fundamental field or state variable.
These are precisely the propositions that give ToE its unifying ambition.
The resulting conceptual architecture can be represented as
$$
\boxed{
\text{information}
\rightarrow
\text{entropy}
\rightarrow
\text{geometry}
\rightarrow
\text{action}
\rightarrow
\text{dynamics}
}
$$
where the arrows indicate proposed explanatory dependence rather than merely chronological processes.
---
10. Entropic Flow as the Dynamical Principle
If the entropic field $S(x)$ is fundamental, then physical evolution can be interpreted as the evolution of systems through an entropic landscape.
The local rate of entropic change is represented by
$$
\frac{dS}{d\tau}
u^\mu\partial_\mu S.
$$
The associated entropic action is then constructed from the integrated entropic evolution along the worldline.
In this picture, physical motion is not conceptually separate from information flow. Motion represents the trajectory through an informationally structured manifold, while the dynamical law determines which trajectories are admissible or extremal.
The phrase entropic flow therefore refers not simply to ordinary thermodynamic heat flow. It refers, within ToE, to the more general evolution of an informational field and its associated geometry.
This distinction is necessary because thermodynamic entropy and the proposed fundamental entropic field $S(x)$ cannot simply be assumed to be identical. ToE must ultimately establish the mathematical relationship between them. If successful, ordinary thermodynamic entropy could emerge as a macroscopic or statistical manifestation of the more fundamental entropic field.
The proposed hierarchy would then be
$$
S_{\mathrm{fundamental}}
\rightarrow
S_{\mathrm{statistical}}
\rightarrow
S_{\mathrm{thermodynamic}},
$$
rather than treating thermodynamic entropy as the primitive quantity.
This distinction also allows ToE to address a central conceptual problem: the word "entropy" is used in several different mathematical senses across physics. Thermodynamic entropy, Gibbs entropy, Shannon entropy, von Neumann entropy, relative entropy, and information-geometric quantities are not automatically identical. A complete ToE must therefore specify the mathematical identity of $S$, its units or normalization, its state space, and its relationship to these established quantities.
---
11. Consequences for Classical Mechanics
The Haller–Obidi Correspondence potentially changes the interpretation of the principal structures of classical mechanics.
The trajectory $q(t)$, ordinarily obtained from
$$
\delta\mathcal{A}=0,
$$
can instead be viewed as an extremal trajectory of the corresponding entropy functional.
The momentum
$$
p_i
\frac{\partial\mathcal{L}}{\partial\dot q^i}
$$
would then acquire an interpretation as a derivative of an entropically defined dynamical functional, depending on the precise Legendre structure adopted by ToE.
Similarly, the Hamiltonian formulation
$$
H_{\mathrm{cl}}
p_i\dot q^i-\mathcal{L}
$$
could potentially be reconstructed from the entropic formalism rather than introduced independently.
The long-term objective is therefore a complete reformulation in which the familiar Lagrangian and Hamiltonian structures emerge as representations of a deeper entropic dynamics.
This would make the correspondence much stronger than a reinterpretation of terminology. It would constitute a derivation of classical mechanics from the entropic formalism.
The desired chain would then be
$$
\mathcal{S}{\mathrm{ent}}
\rightarrow
\mathcal{L}{\mathrm{ent}}
\rightarrow
\mathcal{A}_{\mathrm{ent}}
\rightarrow
\text{Euler--Lagrange equations}
\rightarrow
\text{Newtonian or relativistic dynamics}.
$$
The corresponding Hamiltonian structure would need to follow from a well-defined Legendre transformation. If the entropic theory modifies the canonical structure itself, then the appropriate generalized momentum and Hamiltonian must likewise be derived rather than assumed.
---
12. Extension Toward Relativity and Geometry
The introduction of the four-vector $u^\mu$ and the covariant derivative $\partial_\mu S$ naturally places the construction in a relativistic setting.
The scalar quantity
$$
u^\mu\partial_\mu S
$$
is invariant under coordinate transformations when $S$ is a scalar field and $u^\mu$ is a properly defined four-vector. This provides a natural covariant language for describing entropic evolution along worldlines.
The next theoretical question is substantially deeper: whether the spacetime metric itself can be derived from the entropic field or from the information geometry associated with it.
If the metric can be represented schematically as
$$
g_{\mu\nu}
g_{\mu\nu}[S,\partial S,\partial^2S,\ldots],
$$
then geometry would no longer be an independent background structure. It would become a functional of the underlying entropic field.
At that point, the conceptual chain becomes
$$
S
\rightarrow
g_{\mu\nu}
\rightarrow
\Gamma^\rho_{\mu\nu}
\rightarrow
R^\rho{}_{\sigma\mu\nu}
\rightarrow
\text{gravitational dynamics}.
$$
Such a construction would provide a possible route toward the ToE objective of treating gravity, geometry, and entropy within a common mathematical framework.
However, this step requires explicit field equations. The mere existence of an entropic scalar does not by itself imply that the spacetime metric is emergent from it. That conclusion must be derived.
A complete construction would therefore need to establish the map
$$
S
\mapsto
g_{\mu\nu},
$$
and then demonstrate that the resulting curvature dynamics reproduce, approximate, or generalize the appropriate gravitational field equations.
If the metric is instead an independent dynamical field coupled to $S$, the theory would have a different architecture:
$$
S
\leftrightarrow
g_{\mu\nu}.
$$
The distinction between emergent geometry and coupled geometry is therefore an important open structural question within the development of ToE.
---
13. Extension Toward Quantum Physics
The proposed action–entropy correspondence also creates a possible bridge to quantum mechanics because the classical action already appears fundamentally in quantum phase.
The semiclassical phase is conventionally associated with
$$
\exp\left(\frac{i\mathcal{A}}{\hbar}\right).
$$
If the action is related to entropy through
$$
\mathcal{A}
mc^2t
\frac{\hbar}{2}H,
$$
then formally,
$$
\exp\left(\frac{i\mathcal{A}}{\hbar}\right)
\exp\left(\frac{imc^2t}{\hbar}\right)
\exp\left(-\frac{i}{2}H\right).
$$
This observation does not, by itself, constitute a quantum theory of entropy. Rather, it identifies a mathematically interesting interface between action, phase, and information.
A complete ToE treatment would need to establish how the entropic field produces quantum amplitudes, interference, probability, or quantum-state geometry. Nevertheless, the action–entropy correspondence provides a natural point from which such questions can be investigated.
This becomes especially significant if ToE subsequently connects the entropic field to information-geometric structures such as the Fisher–Rao metric, quantum Fisher information, or Fubini–Study geometry.
The ultimate objective would be to demonstrate that the classical and quantum descriptions are not unrelated mathematical frameworks but limiting manifestations of a common informational geometry.
In that prospective framework, the classical action would represent one projection of the underlying entropic structure, while the quantum phase would provide another. A complete derivation would have to determine whether the relationship is merely formal or whether a genuine quantum dynamical equation follows from the same underlying entropic action.
---
14. The Ontological Reversal Proposed by ToE
The deepest aspect of the Haller–Obidi Correspondence is therefore an ontological reversal.
Conventional physical reasoning often proceeds approximately as
$$
\text{matter}
\rightarrow
\text{fields}
\rightarrow
\text{interactions}
\rightarrow
\text{geometry}
\rightarrow
\text{thermodynamic information}.
$$
ToE investigates the possibility of reversing this explanatory direction:
$$
\text{information}
\rightarrow
\text{entropy}
\rightarrow
\text{geometry}
\rightarrow
\text{dynamics}
\rightarrow
\text{matter}.
$$
In this picture, matter is not necessarily the fundamental substrate from which information is derived. Instead, matter may be a stable manifestation of an underlying informational and entropic structure.
This is particularly relevant to the broader ToE program in which mass is investigated as an emergent quantity associated with geometric or informational moments. If the mass of a physical excitation can ultimately be derived from the geometry of the entropic field rather than introduced as an independent primitive, then the Haller–Obidi Correspondence becomes part of a much larger chain of emergence.
Schematically,
$$
\text{entropic field}
\rightarrow
\text{informational geometry}
\rightarrow
\text{geometric moments}
\rightarrow
\text{mass-energy}
\rightarrow
\text{physical dynamics}.
$$
The Haller–Obidi Correspondence would then provide the action principle connecting that emergent structure to actual physical evolution.
This is a particularly important conceptual point for ToE. If mass, energy, geometry, and dynamics can all be represented as different mathematical manifestations of the same underlying informational structure, then the theory would not merely unify existing equations. It would propose a new hierarchy of physical explanation.
---
15. The Principle of Extremal Entropy
The central variational proposition of the Haller–Obidi framework can therefore be expressed as
$$
\delta\mathcal{S}_{\mathrm{ent}}=0,
$$
where $\mathcal{S}_{\mathrm{ent}}$ denotes the appropriate entropic action or entropy functional.
The corresponding physical equations would follow from
$$
\frac{\delta\mathcal{S}_{\mathrm{ent}}}
{\delta\Phi^A}
0,
$$
where $\Phi^A$ represents the complete set of dynamical fields of the theory.
For a theory containing the entropic scalar $S$, one would have, schematically,
$$
\frac{\delta\mathcal{S}_{\mathrm{ent}}}
{\delta S}
0.
$$
If the metric is also dynamical,
$$
\frac{\delta\mathcal{S}{\mathrm{ent}}}
{\delta g{\mu\nu}}
0.
$$
If matter fields $\psi$ are present,
$$
\frac{\delta\mathcal{S}_{\mathrm{ent}}}
{\delta\psi}
0.
$$
A mature ToE would therefore not merely state that entropy governs physics. It would provide a single action or generalized variational functional whose simultaneous variation generates the relevant field equations.
That is the mathematical threshold separating a philosophical interpretation from a complete field theory.
More generally, one can represent the proposed ToE action as
$$
\mathcal{S}_{\mathrm{ToE}}
\int_{\mathcal{M}}
\mathcal{L}_{\mathrm{ToE}}
\sqrt{|g|},d^n x,
$$
where $\mathcal{M}$ denotes the relevant manifold and $\mathcal{L}_{\mathrm{ToE}}$ contains the entropic, geometric, informational, and matter contributions.
The fundamental variational statement would then be
$$
\delta\mathcal{S}_{\mathrm{ToE}}=0.
$$
The complete field equations would follow from
$$
\frac{\delta\mathcal{S}_{\mathrm{ToE}}}
{\delta\Phi^A}
0.
$$
This provides a natural formal target for the continued development of the Theory of Entropicity.
---
16. What Makes the Haller–Obidi Correspondence Distinctive
The distinctive feature of Obidi's approach is therefore not simply that he invokes entropy in discussing physics. Entropy already appears throughout statistical mechanics, thermodynamics, black-hole physics, information theory, and several approaches to gravitational and quantum foundations.
The distinctive proposal is the attempt to reorganize the hierarchy of physical concepts around entropy and information.
The Haller relation provides the initial bridge:
$$
\text{action}
\longleftrightarrow
\text{entropy}.
$$
Obidi then seeks to extend that bridge:
$$
\text{action}
\longleftrightarrow
\text{entropy}
\longleftrightarrow
\text{field}
\longleftrightarrow
\text{geometry}
\longleftrightarrow
\text{matter}.
$$
This is why the Haller–Obidi Correspondence is central to ToE. It provides a possible route by which the variational machinery of classical physics can be absorbed into a more general informational ontology.
The proposal is consequently not merely that "entropy is important in physics." The stronger proposition is that entropy may constitute part of the mathematical foundation from which physical dynamics themselves arise.
This distinction should remain explicit throughout the ToE monograph. The framework is not claiming that conventional physics is wrong merely because it employs action, energy, mass, geometry, or fields as fundamental or effective quantities. Rather, it proposes that these quantities may themselves admit a deeper interpretation within an entropic and informational hierarchy.
---
17. The Research Program Rather Than a Completed Claim
The Haller–Obidi Correspondence should ultimately be understood as part of an ongoing theoretical construction.
The fundamental insight is the proposed relationship between action and entropy. The next task is to determine the complete mathematical structure that follows from taking that relationship seriously.
That research necessarily includes determining the appropriate field variables, constructing a nontrivial entropic action, deriving the Euler–Lagrange or covariant field equations, identifying conserved quantities through the relevant symmetries, establishing the correct dimensional and physical normalization of $S$, recovering established classical and relativistic limits, and determining whether quantum and gravitational phenomena can emerge from the same formalism.
The fact that the final mathematical structure may not yet be completely resolved does not invalidate the research direction. A theoretical program can legitimately begin with a structural insight and proceed by systematically deriving its consequences.
The governing methodological principle is therefore
$$
\boxed{
\text{follow the mathematics to its logical conclusion}.
}
$$
Obidi's objective is not primarily to make the developing theory conform to existing academic consensus. His objective is to determine whether the mathematical structure generated by the underlying insight can be made internally consistent, physically meaningful, and ultimately empirically testable.
This distinction is fundamental.
Independence from consensus does not mean independence from mathematical rigor. On the contrary, the more unconventional the hypothesis, the more important mathematical discipline becomes.
Thus,
$$
\boxed{
\text{independence from consensus}
\neq
\text{independence from verification}.
}
$$
The theory must ultimately be constrained by mathematics, dimensional consistency, covariance where appropriate, conservation laws, limiting behavior, and experimental evidence.
The research posture can therefore be summarized as
$$
\text{insight}
\rightarrow
\text{mathematical formulation}
\rightarrow
\text{derivation}
\rightarrow
\text{internal consistency}
\rightarrow
\text{physical interpretation}
\rightarrow
\text{prediction}
\rightarrow
\text{verification}.
$$
Academic consensus may be consulted, compared, challenged, or ultimately supported by the resulting theory, but it is not the mathematical principle that determines where the investigation must terminate.
---
18. The Broader ToE Architecture
Within the broader Theory of Entropicity, the Haller–Obidi Correspondence can therefore be viewed as one component of a much larger architecture.
The proposed conceptual progression is
$$
\boxed{
\text{Information}
\rightarrow
\text{Entropy}
\rightarrow
\text{Entropic Field}
\rightarrow
\text{Information Geometry}
\rightarrow
\text{Action}
\rightarrow
\text{Dynamics}
\rightarrow
\text{Matter and Mass}
\rightarrow
\text{Observable Physics}.
}
$$
The Haller contribution occupies the critical action–entropy interface:
$$
\boxed{
\text{Entropy}
\longleftrightarrow
\text{Action}.
}
$$
Obidi's contribution is to ask what happens when that correspondence is promoted into a field-theoretic and geometric principle.
The resulting program potentially connects the variational principles of mechanics with information geometry, thermodynamics, relativistic field theory, quantum theory, and the emergence of physical quantities from geometric structure.
The ambition is therefore not merely to reinterpret individual equations. It is to determine whether the apparently separate mathematical languages of physics can be generated from a common informational foundation.
This larger architecture can be represented schematically as
$$
\text{Information}
\rightarrow
\text{Entropy}
\rightarrow
\text{Entropic Field}
\rightarrow
\text{Informational Geometry}
\rightarrow
\text{Spacetime Geometry}
\rightarrow
\text{Action}
\rightarrow
\text{Field Dynamics}
\rightarrow
\text{Mass and Energy}
\rightarrow
\text{Matter}
\rightarrow
\text{Observable Phenomena}.
$$
The Haller–Obidi Correspondence is therefore not an isolated identity. It is proposed as a structural junction connecting the informational foundation to the dynamical formalism.
---
19. The Relationship Between Entropy and Geometry
Because the broader ToE architecture places information geometry between entropy and physical dynamics, the Haller–Obidi Correspondence naturally leads to a deeper geometric question.
If $S$ represents an entropic state variable, then variations in $S$ may define a structure on an appropriate state space. A generalized information metric might be represented schematically by
$$
g_{ab}^{(S)}
g_{ab}
\left(
S,
\partial_a S,
\partial_b S,
\ldots
\right).
$$
Alternatively, if the theory is formulated directly on a statistical manifold with coordinates $\theta^a$, one could consider an information metric of the form
$$
g_{ab}
\mathbb{E}
\left[
\partial_a \log p(x|\theta)
,
\partial_b \log p(x|\theta)
\right].
$$
In a classical statistical setting, this has the Fisher–Rao form. In quantum settings, related information-geometric structures include the quantum Fisher information and the Fubini–Study metric for pure states.
The ToE research program can therefore investigate whether the entropic field $S$ provides the underlying scalar or field-theoretic structure from which such geometric quantities arise.
The important conceptual transition is
$$
\text{entropy}
\rightarrow
\text{information metric}
\rightarrow
\text{geodesic structure}
\rightarrow
\text{physical dynamics}.
$$
If such a derivation can be established, then geometry would acquire an informational origin rather than being merely imposed as an independent mathematical background.
---
20. Entropy as a Candidate Dynamical Ontology
The deepest claim of the ToE program can consequently be expressed in terms of ontology.
In conventional formulations, entropy is often introduced as a property of a physical state. ToE investigates whether the relationship can be reversed:
$$
\text{physical state}
\leftarrow
\text{entropic structure}.
$$
The distinction is substantial.
If entropy is merely a derived property, then the fundamental variables are physical entities from which entropy is calculated.
If entropy is fundamental, then physical entities may instead emerge as stable structures of the entropic field.
The corresponding ontological hierarchy becomes
$$
\text{entropic information}
\rightarrow
\text{geometry}
\rightarrow
\text{stable configurations}
\rightarrow
\text{physical observables}.
$$
Under this interpretation, particles, fields, mass, energy, and even spacetime structure could potentially be regarded as emergent manifestations of an underlying entropic geometry.
This is the philosophical depth of Obidi's move. It is not merely a substitution of one word for another. It is an attempt to reverse the direction of explanation in fundamental physics.
---
21. The Haller–Obidi Correspondence as a Bridge Between Descriptions
The correspondence can therefore be interpreted at three distinct levels.
At the first level, there is the algebraic correspondence between the classical action and the entropy-like quantity:
$$
H
\frac{2}{\hbar}
\int
\left(
mc^2-\mathcal{L}
\right),dt.
$$
At the second level, there is the variational correspondence:
$$
\delta\mathcal{A}=0
\quad\Longleftrightarrow\quad
\delta H=0,
$$
under the conditions necessary for the action–entropy transformation.
At the third level, there is the field-theoretic correspondence, in which the entropic field $S(x)$ is incorporated into the local dynamical description:
$$
\mathcal{L}_{\mathrm{ent}}
mc^2
\frac{\hbar}{2}
u^\mu\partial_\mu S.
$$
These three levels should not be conflated.
The first establishes a relationship between quantities.
The second establishes a relationship between variational principles.
The third attempts to establish a new dynamical framework.
The third is therefore the most ambitious and requires the greatest amount of mathematical development.
---
22. The Logical Development of the ToE Program
The research logic can be represented as a sequence of increasingly strong propositions.
The first proposition is that action and entropy admit a mathematical correspondence:
$$
\mathcal{A}
\leftrightarrow
H.
$$
The second proposition is that the stationary-action principle can therefore be reformulated as an extremal-entropy principle:
$$
\delta\mathcal{A}=0
\quad\Longleftrightarrow\quad
\delta H=0.
$$
The third proposition is that entropy can be represented by a field:
$$
H
\rightarrow
S(x).
$$
The fourth proposition is that the field enters local dynamics through its directional derivative:
$$
S(x)
\rightarrow
u^\mu\partial_\mu S.
$$
The fifth proposition is that this quantity contributes to an entropic Lagrangian:
$$
u^\mu\partial_\mu S
\rightarrow
\mathcal{L}_{\mathrm{ent}}.
$$
The sixth proposition is that the resulting entropic action generates the physical equations of motion:
$$
\mathcal{L}{\mathrm{ent}}
\rightarrow
\mathcal{A}{\mathrm{ToE}}
\rightarrow
\frac{\delta\mathcal{A}_{\mathrm{ToE}}}{\delta\Phi^A}=0.
$$
The seventh and most ambitious proposition is that the resulting dynamics reproduce or generalize the established physical theories in their appropriate limits:
$$
\mathcal{T}{\mathrm{ToE}}
\rightarrow
\left{
\mathcal{T}{\mathrm{classical}},
\mathcal{T}{\mathrm{relativistic}},
\mathcal{T}{\mathrm{quantum}},
\mathcal{T}_{\mathrm{thermodynamic}},
\ldots
\right}.
$$
This sequence makes clear where the established starting point ends and where the new theoretical construction begins.
---
23. What Must Ultimately Be Resolved
For the Haller–Obidi Correspondence to mature into a complete component of the Theory of Entropicity, several mathematical questions must ultimately be resolved.
The first is the exact definition of the entropy variable $S$. It must be clear whether $S$ is dimensionless, has thermodynamic units, is measured in natural-information units, or is normalized by $\hbar$ or another physical constant.
The second is the exact relationship between $H$ and $S$. If $H$ is a global or trajectory-integrated quantity while $S(x)$ is a local field, the map
$$
H
\longleftrightarrow
S(x)
$$
must be explicitly defined.
The third is the complete entropic action. The worldline expression
$$
\mathcal{A}_{\mathrm{OH}}
\int
\left[
mc^2
\frac{\hbar}{2}
u^\mu\partial_\mu S
\right]d\tau
$$
must be embedded into a formulation in which the entropy field possesses genuine independent dynamics if that is intended.
The fourth is the coupling to geometry. The theory must establish whether
$$
S\rightarrow g_{\mu\nu},
$$
or whether $S$ and $g_{\mu\nu}$ are independent but dynamically coupled.
The fifth is the recovery of known physics. The ToE equations must reproduce appropriate classical, relativistic, thermodynamic, and quantum limits.
The sixth is the derivation of conservation laws. If the theory is variational, the symmetries of the complete action should determine the corresponding conserved quantities through the appropriate generalization of Noether's theorem.
The seventh is empirical distinguishability. A theory becomes physically consequential when it predicts something that can be tested and that distinguishes it from existing descriptions.
These questions do not diminish the original insight. They define the mathematical pathway by which the insight can be developed into a mature theory.
---
24. Conclusion: From Classical Action to Entropic Physics
The Haller–Obidi Correspondence represents a proposed conceptual transition from classical action as a foundational dynamical quantity to entropy as a possible underlying variational quantity.
Beginning with
$$
H
\frac{2}{\hbar}
\int
\left(
mc^2-\mathcal{L}
\right)dt,
$$
the correspondence establishes a structural relationship between entropy and classical action. Obidi's extension introduces an entropic field $S(x)$ and seeks to incorporate its local gradient directly into the Lagrangian:
$$
\mathcal{L}_{\mathrm{ent}}
mc^2
\frac{\hbar}{2}
u^\mu\partial_\mu S.
$$
The corresponding action,
$$
\mathcal{A}_{\mathrm{OH}}
\int
\mathcal{L}_{\mathrm{ent}},d\tau,
$$
provides the proposed foundation for an entropic reformulation of dynamics.
The central conceptual transformation can be summarized as
$$
\text{stationary action}
\quad\longrightarrow\quad
\text{extremal entropy}.
$$
From this starting point, ToE investigates the possibility that entropy is not merely a thermodynamic descriptor but a deeper field-theoretic quantity whose geometry determines physical evolution.
The broader philosophical proposition is consequently
$$
\text{information}
\rightarrow
\text{entropy}
\rightarrow
\text{geometry}
\rightarrow
\text{action}
\rightarrow
\text{dynamics}
\rightarrow
\text{physical reality}.
$$
If this program can be completed mathematically, then classical mechanics would not be discarded but recovered as an effective sector of a deeper entropic theory. Relativistic dynamics could emerge from the covariant geometry of the entropic field. Quantum behavior could potentially arise through the relationship between entropic action and quantum phase. Mass and energy could potentially emerge from deeper informational-geometric structures. Thermodynamics could become a macroscopic manifestation of the same underlying entropic field.
The ultimate significance of the Haller–Obidi Correspondence therefore lies in the possibility that the classical action principle is not the terminus of the explanation of physical dynamics but a visible manifestation of a deeper informational variational principle.
In its most concise form, the ToE proposition is:
$$
\text{Physics evolves according to the extremal structure of information and entropy.}
$$
The Haller–Obidi Correspondence is the proposed mathematical bridge through which that proposition begins to connect with the established variational language of physics.
Its final significance, however, will depend on whether the developing ToE formalism can carry this correspondence beyond reinterpretation and into a complete dynamical theory: one possessing well-defined fields, a nontrivial action, rigorous equations of motion, recoverable physical limits, conserved quantities, quantitative predictions, and ultimately empirical tests.
That is the research trajectory opened by the correspondence. Obidi's task is not to stop at the first elegant identity, nor to force the emerging theory into the shape of existing consensus, but to continue following the mathematical structure wherever it leads until the final form of the underlying theory is resolved.
The central methodological principle can therefore be stated one final time:
$$
\text{follow the mathematics to its logical conclusion}.
$$
The purpose of the Theory of Entropicity is not to obtain agreement with an existing conceptual framework merely for the sake of agreement. Its purpose is to determine whether the proposed entropic structure can withstand the full sequence of mathematical and physical demands imposed upon a fundamental theory.
Thus the ultimate standard is neither novelty alone nor conformity alone. It is the convergence of mathematical coherence, physical meaning, limiting consistency, predictive power, and empirical verification:
$$
\boxed{
\text{Insight}
\rightarrow
\text{Mathematics}
\rightarrow
\text{Dynamics}
\rightarrow
\text{Prediction}
\rightarrow
\text{Verification}.
}
$$
Within that methodological framework, the Haller–Obidi Correspondence represents not the conclusion of the Theory of Entropicity, but a potentially foundational bridge from which the deeper structure of the theory can continue to be derived.
