The Theory of Entropicity posits that entropy is the fundamental substrate of reality, driving all physical processes and giving rise to spacetime, forces, and quantum phenomena.
In standard quantum mechanics, a system is described by a wavefunction
\psi
whose evolution is governed by the Schrödinger equation
i\hbar \frac{\partial \psi}{\partial t} = \hat{H}\psi .
This equation is continuous and deterministic.
However, when a measurement occurs, the result is discrete. For example:
The transition from the continuous wavefunction to discrete outcomes is the famous measurement problem.
Traditional interpretations introduce additional postulates such as wavefunction collapse, decoherence, or branching universes. None of these explanations derive discreteness from deeper physical principles.
This is where the Theory of Entropicity introduces a new possibility.
At its most basic level, measurement is the creation of a distinguishable state.
Before measurement:
the system is described by a superposition
|\psi\rangle = \sum_i c_i |i\rangle .
After measurement:
the system occupies one of the eigenstates
|i\rangle .
But what does it actually mean for a measurement outcome to be “real”?
It means the state of the system has become physically distinguishable from other possible states.
Thus measurement is fundamentally a process of state distinguishability.
In information geometry, distinguishability between states is measured by relative entropy.
For two distributions and :
D_{KL}(P||Q) =
\sum_i P_i \ln \frac{P_i}{Q_i}.
In quantum theory the analogous quantity is quantum relative entropy.
These quantities measure how distinguishable two states are.
But they do not specify a minimum threshold for physical distinguishability.
They allow arbitrarily small separations.
The Theory of Entropicity introduces precisely the missing ingredient.
It proposes that the entropic field possesses a minimum curvature gap:
\mathcal{C}_{OCI} = \ln 2.
This is the smallest entropic separation that produces a physically distinguishable configuration.
Below this threshold, differences exist mathematically but not physically.
Thus the OCI acts as a distinguishability threshold.
Measurement can now be reinterpreted.
During a measurement interaction, the entropic field describing the system and apparatus evolves continuously.
However, the measurement result becomes physically real only when the entropic separation between alternative outcomes exceeds
\Delta S = k_B \ln 2.
At that moment, the configurations become physically distinguishable states of the entropic field.
Thus measurement outcomes correspond to entropic states separated by the OCI curvature gap.
This framework explains a major mystery of quantum theory.
The entropic field evolves continuously, but distinguishable states appear only after crossing the OCI threshold.
Therefore:
continuous evolution
→ threshold crossing
→ discrete outcomes
The discreteness of measurement results emerges naturally from the geometry of the entropic manifold.
This mechanism resembles a phase transition in distinguishability.
Landauer's principle states that erasing one bit of information requires
\Delta E = k_B T \ln 2 .
Within the entropic field framework this has a deeper interpretation.
The energy corresponds to the work required to deform the entropic field enough to cross the OCI threshold.
Thus the thermodynamic cost of information processing reflects the energetic cost of creating a distinguishable state.
Measurement, information, and entropy become different manifestations of the same entropic geometry.
Under this interpretation, wavefunction collapse is not a mysterious discontinuous process.
Instead, the wavefunction describes the continuous evolution of entropic configurations.
Collapse occurs when the entropic separation between possible outcomes exceeds the OCI threshold.
At that moment, the states become distinguishable and one branch becomes physically realized.
Thus collapse becomes a geometric event in the entropic manifold.
Because the smallest distinguishable separation corresponds to
\Delta S = k_B \ln 2,
the simplest measurement outcome corresponds to a binary distinction.
This explains why measurement apparatus ultimately produce binary signals:
detector click / no click
spin up / spin down
0 / 1
Binary information arises naturally because the smallest entropic curvature separation corresponds to two distinguishable states.
The connection above suggests a principle that we must now formulate formally within the Theory of Entropicity (ToE):
Entropic Measurement Principle
A quantum measurement outcome becomes physically realized when the entropic separation between alternative system–apparatus configurations exceeds the Obidi Curvature Invariant threshold (OCI).
This principle replaces the abstract notion of wavefunction collapse of Quantum Mechanics with a geometric criterion based on entropy.
If this ToE interpretation holds, it connects several areas of physics that have traditionally been separate:
The constant , which already appears in all these domains, becomes the universal distinguishability constant of physical reality.
This is why this ToE connection is potentially powerful.
It suggests that the discreteness of quantum measurement may arise from a geometric property of the entropic field rather than from an additional postulate of quantum theory.
This ToE idea is interesting because it reframes something physicists already know.
Everyone knows that
k_B \ln 2
appears in:
But those appearances are usually treated as separate facts.
The Theory of Entropicity proposes that they reflect one underlying physical structure: the minimum curvature required for distinguishability in the entropic field.
If this ToE interpretation is correct, then the constant is not merely a unit conversion factor but a geometric invariant governing how physical states become distinguishable.
This connection between the Obidi Curvature Invariant and quantum measurement of Quantum Mechanics does not claim to solve the measurement problem outright. However, it suggests a possible mechanism by which the discrete outcomes of measurement may arise from the geometry of entropy itself.
If entropy is indeed the fundamental field underlying physical reality, then measurement may represent the moment when the entropic manifold differentiates sufficiently for one configuration to become physically distinguishable from the others.
In this view, quantum measurement is not a mysterious discontinuity in the laws of physics. Rather, it is the natural consequence of the geometry of the entropic field of Obidi's Theory of Entropicity (ToE).
The Theory of Entropicity (ToE) has been able to construct physical Riemannian spacetime (RS) from information geometry (IG) by promoting information geometry from a statistical descriptor to an ontological field geometry. That promotion is the distinctive move and achievement of the Theory of Entropicity (ToE). The underlying mathematical ingredients—Fisher–Rao metrics, Fubini–Study metrics, α-connections, emergent-metric programs, and entropy-based gravity—already exist in the literature【Obidi 2025/2026; Jacobson 1995; Verlinde 2010—2011; Bianconi 2021–2025 as referenced in Obidi’s foundations papers].
What ToE posits is that these are not merely useful formalisms but partial shadows of one deeper entropic manifold.
In standard information geometry, one starts with a family of probability distributions or quantum states and equips that family with a metric. In the classical case this is typically the Fisher–Rao metric; in the quantum case, one encounters the Fubini–Study metric or related monotone metrics. These are already bona fide Riemannian metrics, but they are usually interpreted as metrics on state space, not on physical spacetime itself. They tell us how distinguishable states are, not where matter lives or how rulers measure distances in the external [physical] world.
The Theory of Entropicity (ToE) changes the status of that geometry completely. It begins with the single axiom that entropy is a universal physical field. Once this is accepted, the manifold of entropic configurations is no longer epistemic. It becomes physical. Then the information metric is no longer merely a metric of inference; it becomes the seed from which physical geometry can emerge.
Formally, the move which the Theory of Entropicity (ToE) has made looks like this. Let the local entropic configuration be parameterized by coordinates on a statistical or informational manifold. The Fisher–Rao metric is:
X
In ordinary information geometry, this is the metric on the space of distributions . In ToE, one interprets those distributions not as subjective probabilities but as local entropic density profiles determined by the field. Then becomes an induced metric on the entropic manifold itself.
Similarly, in a quantum sector one may write the Fubini–Study metric on projective Hilbert space as:
X
Again, standard theory treats this as geometry of quantum states. But the Theory of Entropicity (ToE) goes one major step further to posit that this, too, is an emergent slice of the deeper entropic geometry. In the ToE literature, the α-connection is used by John Onimisi Obidi as the bridge that unifies the Fisher–Rao and Fubini–Study sectors into a single entropic-geometric framework. That is where the construction becomes specifically ToE native.
The next step in the revolutionary insight and trajectory of ToE is the crucial one: how does one get from such information geometry to physical spacetime?
ToE’s answer and resolution of this impasse is to declare that [physical] spacetime [itself] is an emergent effective metric induced by the entropic field. The information metric is first defined on the configuration manifold of the entropic field; then, through the Obidi Action and the Master Entropic Equation (MEE) — otherwise known as the Obidi Field Equations (OFE), one identifies the effective spacetime metric as a functional of S, its gradients, and its information-geometric invariants. In schematic form, the ToE move is given as:
X1
where X2 is the information-geometric curvature scalar. In the simplest versions of Obidi's ToE program, the physical metric is induced from the Levi–Civita slice of the entropic information geometry, which is why the ToE papers repeatedly place Fisher–Rao, Fubini–Study, and Amari–Čencov structures inside one emergent-geometric chain.
So, the ToE construction is not “information geometry somehow magically becomes spacetime.” The rigorous claim that ToE is making is even much more strict and more defensible:
That is the formal route and core foundation of Obidi's Theory of Entropicity (ToE).
Now, let us turn to the second part of our inquiry: is this unique to ToE?
We acknowledge that the attempt to derive spacetime from information or entropy is not unique to the Theory of Entropicity (ToE), at least not speaking at the level of the general and rather broad and audacious ambition that ToE has undertaken. However, many researchers have tried to derive spacetime or gravity from thermodynamics, information, or entanglement. Ted Jacobson derived the Einstein field equations from Clausius-type thermodynamic reasoning. Erik Verlinde proposed entropic gravity. Ginestra Bianconi constructed gravity-from-entropy programs using information geometry and metric relative entropy. There are also quantum-information and tensor-network programs in which geometry emerges from entanglement or distinguishability. So the broad project “physical geometry from informational structure” is not unique to ToE
What is more plausibly and undoubtedly unique to ToE is the specific ontological and structural synthesis, which we must now address:
First, ToE does not merely say information is useful for describing geometry. It says entropy is the fundamental field of reality. That is stronger than Jacobson, Verlinde, Bianconi or most information-geometric programs.
Second, ToE embarks on a bold, courageous and at the same time intimidating trajectory to unify classical and quantum information geometry through the α-connection within one physical field picture, rather than leaving Fisher–Rao and Fubini–Study as separate mathematical domains.
Third, ToE introduces the Obidi Curvature Invariant (OCI) as a threshold of distinguishability and uses it to regulate when physical geometry and events become realized. That threshold structure is not part of the standard emergent-spacetime literature as such, and is unique to ToE.
Fourth, ToE combines this [Obidi Curvature Invariant (OCI)] with the No-Rush Theorem (NRT) and No-Go Theorem (NGT) frameworks of ToE, so that emergent spacetime is not just geometric but thresholded and temporally constrained in its physical realization.
Hence, we can conclude as follows:
The construction of spacetime from information geometry is not unique to ToE as a research direction. What is distinctive and irrefutably unique in ToE is that information geometry is not treated as a mathematical analogy or derived description, but as the physical geometry of a universal entropic field from which Riemannian spacetime is induced.
That is the strong and defensible claim of Obidi's Theory of Entropicity (ToE).
There is one more important qualification that is crucial for us to make on behalf of ToE. For ToE to fully establish this construction in the eyes of mathematical physicists, it still needs an explicit derivation showing, step by step, how a Lorentzian or Riemannian spacetime metric satisfying familiar physical limits emerges from the Obidi Action and the information-geometric sector. The conceptual framework is there. The uniqueness claim is partly there. But the strongest version of the result requires the full derivation. Obidi has already made a brave attempt at this in the available literature, to which we must here refer the reader.
So, we can conclude our program here on Obidi's Theory of Entropicity (ToE) as follows:
The Theory of Entropicity (ToE) has constructed physical spacetime from information geometry by treating entropy as a real field whose local configurations induce a Fisher–Rao / Fubini–Study–type metric, then promoting that information metric to a dynamical physical geometry through the Obidi Action. This is not unique in broad ambition, because other emergent-gravity and information-geometric programs exist, but ToE is irrefutably and undoubtedly distinctive in turning entropy itself into the ontological substrate and in attempting to unify the classical, quantum, and geometric sectors within one entropic field framework—which formalism and methodology are conspicuously absent from all other theories.