How Has the Theory of Entropicity (ToE) Been able to Construct Riemannian Physical Spacetime from Information Geometry, and What is its Uniqueness? 

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:

1. The entropy field defines local entropic state profiles.
2. Those profiles induce an information metric.
3. The Obidi Action makes that information metric dynamical.
4. The smooth, low-energy, macroscopic limit of that dynamical metric is identified with physical Riemannian spacetime.

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.

References

https://entropicity.github.io/Theory-of-Entropicity-ToE/equations/how-toe-derives-physical-spacetime-from-information-geometry-fisher-rao-fubini-study-amari-cencov-levi-civita-riemannian-geometry.html