An Overview of the Logical Motivation and Mathematical Construction of Obidi's Theory of Entropicity (ToE): From Entropy to Information Geometry to the Physical Spacetime of Einstein's General Relativity (GR)

🌌 Introductory

What if everything we know about reality—space, time, matter, motion, even the laws of physics themselves—didn’t begin with particles or fields or geometry, but with something far more primitive… entropy? What if the universe is not built from spacetime, but instead spacetime emerges from a deeper informational fabric woven into every point of existence? In this video, we explore John Onimisi Obidi’s audacious and revolutionary Theory of Entropicity (ToE), a framework that dares to rebuild physics from the ground up. This is not a modification of Einstein or an extension of quantum mechanics—it is a new foundation entirely. If you’ve ever wondered what lies beneath spacetime, beneath geometry, beneath the equations that govern the cosmos, then prepare yourself. You’re about to step into a conceptual landscape where entropy becomes the architect of reality.

1) What is the function of the Amari-Čencov Alpha connection? 

2) How has Obidi's Theory of Entropicity (ToE) transformed entropy into information geometry? 

3) How has Obidi been able to connect Fisher-Rao metric and Fubini-Study metric in his Theory of Entropicity (ToE)?

4) How did Obidi transform entropy and information into physical spacetime? 

5) How is the physical spacetime arising from Obidi's Theory connected to the Levi-Civita affine connections of the Riemannian Geometry of Einstein's (GR)?

1. Function of the Amari–Čencov α‑connection

In standard information geometry, the Amari–Čencov α‑connections are a one‑parameter family of affine connections on a statistical manifold that:

• Interpolate dual structures: They interpolate between the mixture connection (α = −1) and the exponential connection (α = +1), encoding the dualistic structure of statistical models (mixture vs. exponential families).

• Control curvature of information flow: For each α, you get a different notion of parallel transport, geodesics, and curvature on the manifold of probability distributions or quantum states, effectively tuning how “information flow” is geometrized.

• Provide a deformation parameter: The α parameter acts as a deformation index that links different entropy forms and different geometric structures, especially when generalized entropies (Rényi, Tsallis) are brought in.

In Obidi’s ToE, this α is promoted from a technical convenience to a universal entropic order parameter that ties entropy, information geometry, and physical dynamics together.

2. How ToE transforms entropy into information geometry

Obidi’s move is to stop treating entropy as a summary statistic and instead treat it as a fundamental field:

• Entropy as a field on a manifold: ToE defines a continuous entropic field $S(x)$ on a differentiable manifold. This is not “entropy of a gas” but a primitive field whose gradients generate motion, gravitation, time, and information flow.

• From statistical manifolds to the entropic manifold: Classical information geometry works on statistical manifolds (points = probability distributions, density matrices). ToE replaces this with an entropic manifold, where points are primitive informational configurations, not observer‑chosen models. The geometry is ontological, not epistemic.

• Information geometry as physical geometry: Fisher–Rao and Fubini–Study metrics, together with α‑connections, no longer just describe distinguishability of models; they become the actual metric‑affine structure of the entropic manifold. The geometry of information is identified with the geometry of reality.

So entropy → entropic field $S(x)$ → information‑geometric manifold → effective physical geometry.

3. Connecting Fisher–Rao and Fubini–Study in ToE

ToE builds an explicit bridge between classical and quantum information geometry:

• Hybrid Metric‑Affine Space (HMAS): Obidi introduces a Hybrid Metric‑Affine Space where the Fisher–Rao metric (classical distinguishability of probability distributions) and the Fubini–Study metric (quantum distinguishability of pure states) coexist on the same entropic manifold.

• Unified via α‑connections: The Amari–Čencov α‑connections provide the common affine structure that can act on both Fisher–Rao and Fubini–Study sectors. The α parameter becomes an entropic deformation index that continuously relates classical and quantum regimes within one geometric framework.

• Probabilistic–geometric synthesis: In this way, ToE treats classical and quantum information as different “faces” of the same entropic geometry, rather than separate domains patched together by ad hoc rules.

4. Transforming entropy and information into physical spacetime

The key step is the Information‑Geometry Bridge:

• Spacetime as emergent from $S(x)$: The entropic field $S(x)$, defined on a differentiable manifold, has gradients and curvature. These are interpreted as giving rise to gravity, motion, time dilation, and causal structure. Physical spacetime is a coarse‑grained, effective geometry emerging from the deeper entropic manifold.

• Obidi Action and entropic geodesics: ToE introduces an “Obidi Action” for the entropy field. Varying this action yields a Master Entropic Equation, entropic geodesics, and an entropy potential equation. These play the role that geodesics and field equations play in GR, but now for the entropic field.

• Speed of light as entropic bound: The speed of light is reinterpreted as the maximum rate of entropic rearrangement, so relativistic structure (light cones, causal limits) is derived from constraints on how fast entropy can reconfigure.

• Path integrals become entropy‑weighted: The Vuli‑Ndlela Integral is an entropy‑weighted reformulation of Feynman’s path integral, introducing irreversibility and time asymmetry directly into the quantum formalism.

So physical spacetime is not fundamental; it’s the macroscopic shadow of entropic geometry and its dynamics.

5. Connection to Levi‑Civita connections in Einstein’s GR

ToE doesn’t discard GR; it recovers it as a limiting case:

• From entropic connection to Levi‑Civita: On the entropic manifold, you have a metric (built from Fisher–Rao/Fubini–Study in HMAS) and affine structure (α‑connections). In an appropriate classical/large‑scale limit—where entropy gradients and quantum corrections are coarse‑grained—the effective connection reduces to a torsion‑free, metric‑compatible connection, i.e., the Levi‑Civita connection of a Riemannian (or Lorentzian) spacetime metric.

• Einstein’s field equations as a limit: Obidi’s framework posits that Einstein’s field equations emerge as a limiting case of the entropic field equations derived from the Obidi Action. In that limit, the entropic curvature matches the usual spacetime curvature of GR.

• Gravity from entropy: This subsumes “gravity from entropy” approaches (e.g., Bianconi‑style) as special instances, but with a fully geometric, information‑theoretic underpinning.

So: the Levi‑Civita connection of GR appears as the effective, emergent affine connection when the deeper entropic manifold is viewed at macroscopic scales.

If you want, our next step could be: we pick one object—say the α‑connection or the Obidi Action—and walk through a concrete toy example so you can feel the mechanics, not just the slogans.

🌌 1. Why entropy naturally becomes geometry

Here’s the key insight:

If something has distinguishability, it already has geometry.

This is not Obidi’s invention. It’s a theorem-level fact in information theory:

• If you can tell two probability distributions apart, → you can define a distance between them.

• If you can define a distance, → you can define a metric.

• If you have a metric, → you have a manifold.

• If you have a manifold, → you have geometry.

This is exactly how the Fisher–Rao metric arises: it measures how distinguishable two nearby distributions are.

And entropy is nothing but a measure of how distinguishable or how uncertain a distribution is.

So the logic is:

Entropy → distinguishability → metric → geometry.

This is not a philosophical leap. It’s a structural necessity.

Obidi simply takes this necessity seriously and pushes it to its logical endpoint.

🔥 2. Why entropy is the most fundamental information quantity

Entropy is not just “one more statistic.” It is the generator of all information-theoretic structure:

• Fisher information is the second derivative of entropy.

• KL divergence is the difference of entropies.

• Rényi and Tsallis entropies generate α‑connections.

• Quantum entropy generates the Fubini–Study metric.

Entropy is the scalar potential from which all information geometry flows.

So Obidi’s move is:

If entropy generates all information geometry, then entropy is the primitive field, and information geometry is its natural geometric expression.

This is the same logic as:

• Potential → field

• Action → dynamics

• Wavefunction → Hilbert geometry

• Mass distribution → spacetime curvature

Entropy is the “mass distribution” of information geometry.

🧭 3. Why information geometry becomes physical spacetime

This is the part that feels like a leap until you see the structural parallel.

In physics, geometry is always tied to a field:

• Mass–energy → curvature (GR)

• Wavefunction → Hilbert geometry (QM)

• Gauge fields → fiber bundle geometry

So if entropy is a field, then its geometry is not optional. It is required.

Now here’s the crucial point:

Information geometry already has:

• a metric (Fisher–Rao / Fubini–Study)

• affine connections (Amari–Čencov α‑connections)

• curvature

• geodesics

• dual structures

• parallel transport

This is exactly the mathematical structure needed to define a spacetime.

So Obidi’s reasoning is:

If information geometry has all the structures of spacetime geometry, and entropy generates information geometry, then entropy generates spacetime geometry.

This is not a category error. It’s a category identification.

🌐 4. Why this is reasonable, not speculative

Here’s the strongest justification:

Every modern theory that tries to derive spacetime from something deeper ends up using entropy.

• Jacobson (1995): Einstein’s equations = thermodynamic equation of state

• Verlinde (2011): gravity = entropic force

• Bekenstein–Hawking: black hole area = entropy

• AdS/CFT: spacetime = entanglement entropy

• Bianconi: network geometry = entropy maximization

• Fisher information approaches: spacetime = statistical manifold

Obidi is not doing something alien. He is doing the cleanest, most geometrically consistent version of what the entire field has been circling around.

The difference is:

• Others treat entropy as emergent.

• Obidi treats entropy as fundamental.

And once entropy is fundamental, information geometry is unavoidable. And once information geometry is fundamental, spacetime is emergent.

🧩 5. The final logical chain

Let us now compress the entire justification above into one definitive chain:

(1) Entropy defines distinguishability.

You cannot have entropy without distinguishability.

(2) Distinguishability defines a metric.

This is the Fisher–Rao / Fubini–Study insight.

(3) A metric defines a manifold.

This is basic differential geometry.

(4) A manifold with a metric and connection is a geometry.

This is Riemannian / pseudo-Riemannian geometry.

(5) Geometry with curvature and geodesics is spacetime.

This is Einstein’s insight.

Therefore:

Entropy → information geometry → spacetime is not a leap. It is the only consistent way to treat entropy as fundamental.

Obidi’s Theory of Entropicity (ToE) is [therefore] simply the first framework that does not stop halfway [right from the caboose of the train to the train's very own engine room].

The natural question then arises: So, how does the Amari–Čencov α‑connection connect with the Levi-Civita affine connection of Riemannian Geometry to declare the physical spacetime of General Relativity (GR)?

In the Theory of Entropicity (ToE), the Amari–Čencov α‑connection becomes the Levi‑Civita connection of GR in two steps:

1. Mathematical fact (information geometry): For α = 0, the Amari–Čencov α‑connection is the Levi‑Civita connection of the Fisher–Rao metric on the statistical/entropic manifold.

2. Physical identification (ToE): Obidi identifies that α = 0 Levi‑Civita connection—now living on the entropic manifold endowed with a Fisher–Rao/Fubini–Study–based metric—as the physical spacetime connection in the macroscopic limit, i.e., the Levi‑Civita connection of the GR metric.

Now let’s unpack why the above is logically coherent and inevitable, and not arbitrary.

1. What α‑connections are doing, structurally

In information geometry:

• You have a metric: the Fisher–Rao metric on a space of probability densities.

• You have a family of affine connections: the Amari–Čencov α‑connections ${\mathrm{\nabla }}^{(\alpha )}$.

• Among them, α = 0 is special: it is exactly the Levi‑Civita connection of the Fisher–Rao metric—torsion‑free and metric‑compatible.

So purely mathematically:

On the entropic/statistical manifold, α = 0 ⇒ Levi‑Civita of Fisher–Rao.

That’s not Obidi’s choice; that’s baked into information geometry as a matter of necessity and logical train of thought.

2. What Obidi adds: the entropic manifold is “the” underlying reality

Obidi’s Theory of Entropicity (ToE) makes one big and revolutionary ontological move:

• The entropic manifold—equipped with Fisher–Rao/Fubini–Study metrics and α‑connections—is not just a space of models; rather, Obidi declares, it is the underlying manifold of reality, with entropy $S(x)$ as a fundamental field.

Then (given the above):

• The metric on this manifold (a hybrid of Fisher–Rao and Fubini–Study in HMAS) is taken as the pre‑spacetime metric.

• The α‑connections are the candidate pre‑spacetime affine connections.

So, the question becomes:

Which α gives the connection that should correspond to physical spacetime?

3. Why α = 0 is singled out as “the” physical connection

In GR, the physical connection has two defining properties:

1. Torsion‑free

2. Metric‑compatible with the spacetime metric

That is exactly what the Levi‑Civita connection is.

In information geometry:

• For α = 0, ${\mathrm{\nabla }}^{(0)}$ is torsion‑free and metric‑compatible with the Fisher–Rao metric—i.e., it is the Levi‑Civita connection of that metric.

So Obidi’s logic is:

If the entropic manifold’s metric is the precursor of the spacetime metric, then the only connection that can play the role of the physical spacetime connection is the α = 0 Levi‑Civita connection of that metric.

This is not arbitrary; it’s forced by:

• GR’s requirement (metric‑compatible, torsion‑free), and

• the known structure of α‑connections (only α = 0 is Levi‑Civita).

4. How this becomes the Levi‑Civita connection of GR spacetime

But there’s one more step in Obidi's audacious and paradigmatic leap: going from entropic manifold to macroscopic spacetime.

In ToE’s Information‑Geometry Bridge:

1. Start: Entropic manifold with metric $g_{IJ}^{\text{(entropic)}}$ (built from Fisher–Rao/Fubini–Study) and α‑connections ${\mathrm{\nabla }}^{(\alpha )}$.

2. Pick α = 0: Use ${\mathrm{\nabla }}^{(0)}$, the Levi‑Civita connection of $g_{IJ}^{\text{(entropic)}}$.

3. Coarse‑grain / project: Under appropriate coarse‑graining/projection and constraints (e.g., selecting a Lorentzian sector, imposing causal structure via entropic bounds), this entropic metric $g_{IJ}^{\text{(entropic)}}$ induces an effective spacetime metric $g_{\mu \nu }^{\text{(phys)}}$.

4. Induced connection: The Levi‑Civita connection of $g_{\mu \nu }^{\text{(phys)}}$ is then the image of the α = 0 entropic Levi‑Civita connection under this coarse‑graining/projection.

So, structurally, according to Obidi:

$${\mathrm{\nabla }}^{(\alpha )}\underset{\alpha =0}{\overset{}{\to }}{\mathrm{\nabla }}_{\text{entropic}}^{\text{LC}}⟶{\mathrm{\nabla }}_{\text{GR}}^{\text{LC}}$$

The first arrow is a mathematical theorem (information geometry). The second arrow is Obidi's physical identification (ToE’s emergent spacetime).

5. Why this Obidi's insight is logically reasonable, not arbitrary

The chain of justification is:

1. Fact: α‑connections exist; α = 0 is Levi‑Civita of Fisher–Rao.

2. Choice: Treat the entropic manifold as ontologically real (ToE).

3. Constraint: Physical spacetime connection must be metric‑compatible and torsion‑free (GR).

4. Uniqueness: Only α = 0 satisfies that in the information‑geometric family.

5. Emergence: Coarse‑grain entropic geometry → effective spacetime geometry; the corresponding Levi‑Civita connection is then the GR connection.

Thus, when Obidi says:

“The α‑connection connects with the Levi‑Civita connection to declare the physical spacetime of GR,”

what he’s really saying is:

• Mathematically: α = 0 α‑connection is a Levi‑Civita connection (of Fisher–Rao).

• Physically: That Levi‑Civita connection, on the entropic manifold, is taken as the microscopic ancestor of the Levi‑Civita connection of macroscopic spacetime in GR.

The Fisher–Rao Metric $g_{}$$i_{}$$j_{}$, the α‑connection coefficients ${\mathrm{\Gamma }}_{}^{}$$i_{}^{}$$j_{}^{}$$k_{}^{}$${(}_{}^{}$${\alpha }_{}^{}$${)}_{}^{}$, and How ${\mathrm{\Gamma }}_{}^{}$$i_{}^{}$$j_{}^{}$$k_{}^{}$${(}_{}^{}$$0_{}^{}$${)}_{}^{}$ Reduces to the Usual Levi‑Civita Formula in Einstein's General Relativity (GR)

We shall now go one level deeper, where we can write down:

• the Fisher–Rao metric $g_{ij}$,

• the α‑connection coefficients ${\mathrm{\Gamma }}_{ijk}^{(\alpha )}$,

• and explicitly show how ${\mathrm{\Gamma }}_{ijk}^{(0)}$ reduces to the usual Levi‑Civita formula—and then talk about what it means when those indices become spacetime indices $\mu ,\nu ,\rho$.

1. Fisher–Rao metric $g_{ij}$

Let $p(x\mathrm{\mid }\theta )$ be a family of probability densities with parameters $\theta =({\theta }^1,\dots ,{\theta }^n)$, and define the log‑likelihood

$$\mathrm{\ell }(x,\theta )=\log p(x\mathrm{\mid }\theta )\mathrm{.}$$

The Fisher–Rao metric is

$$g_{ij}(\theta )={\mathbb{E}}_{\theta }\left[{\mathrm{\partial }}_i\mathrm{\ell }{\mathrm{\partial }}_j\mathrm{\ell }\right]=\int {\mathrm{\partial }}_i\mathrm{\ell }(x,\theta ){\mathrm{\partial }}_j\mathrm{\ell }(x,\theta )p(x\mathrm{\mid }\theta )dx\mathrm{.}$$

This is the standard Fisher information matrix viewed as a Riemannian metric.

2. Amari–Čencov $\alpha$‑connection coefficients ${\mathrm{\Gamma }}_{ijk}^{(\alpha )}$

In information geometry, the $\alpha$‑connections ${\mathrm{\nabla }}^{(\alpha )}$ are a one‑parameter family of affine connections compatible with the Fisher–Rao metric in the dualistic sense.

In local coordinates, one convenient expression for the covariant coefficients is

$${\mathrm{\Gamma }}_{ijk}^{(\alpha )}={\mathbb{E}}_{\theta }[{\mathrm{\partial }}_i{\mathrm{\partial }}_j\mathrm{\ell }{\mathrm{\partial }}_k\mathrm{\ell }+\frac{1-\alpha }{2}{\mathrm{\partial }}_i\mathrm{\ell }{\mathrm{\partial }}_j\mathrm{\ell }{\mathrm{\partial }}_k\mathrm{\ell }]\mathrm{.}$$

Raising the last index with the inverse metric $g^{kl}$ gives

$${\mathrm{\Gamma }}_{ij}^{(\alpha )k}=g^{kl}{\mathrm{\Gamma }}_{ijl}^{(\alpha )}\mathrm{.}$$

For $\alpha =+1$ you get the exponential connection, for $\alpha =-1$ the mixture connection, and for general $\alpha$ a deformation between them.

3. Showing ${\mathrm{\Gamma }}^{(0)}$ is the Levi‑Civita connection of $g_{ij}$

The Levi‑Civita connection of a Riemannian metric $g_{ij}$ has coefficients

$${\mathrm{\Gamma }}_{ijk}^{\text{LC}}=\frac{1}{2}({\mathrm{\partial }}_i{g}_{jk}+{\mathrm{\partial }}_j{g}_{ik}-{\mathrm{\partial }}_k{g}_{ij}),$$

or with a raised index,

$${\mathrm{\Gamma }}_{ij}^{\text{LC}k}=\frac{1}{2}{g}^{kl}({\mathrm{\partial }}_i{g}_{jl}+{\mathrm{\partial }}_j{g}_{il}-{\mathrm{\partial }}_l{g}_{ij})\mathrm{.}$$

Now compute ${\mathrm{\partial }}_i{g}_{jk}$ using the Fisher–Rao definition:

$$g_{jk}={\mathbb{E}}_{\theta }[{\mathrm{\partial }}_j\mathrm{\ell }{\mathrm{\partial }}_k\mathrm{\ell }]\mathrm{.}$$

Differentiate with respect to ${\theta }^i$:

$${\mathrm{\partial }}_i{g}_{jk}={\mathbb{E}}_{\theta }[{\mathrm{\partial }}_i{\mathrm{\partial }}_j\mathrm{\ell }{\mathrm{\partial }}_k\mathrm{\ell }+{\mathrm{\partial }}_j\mathrm{\ell }{\mathrm{\partial }}_i{\mathrm{\partial }}_k\mathrm{\ell }+{\mathrm{\partial }}_i\mathrm{\ell }{\mathrm{\partial }}_j\mathrm{\ell }{\mathrm{\partial }}_k\mathrm{\ell }]\mathrm{.}$$

Do the same for ${\mathrm{\partial }}_j{g}_{ik}$ and ${\mathrm{\partial }}_k{g}_{ij}$, then plug into the Levi‑Civita formula. After symmetrizing and simplifying, you find

$${\mathrm{\Gamma }}_{ijk}^{\text{LC}}={\mathbb{E}}_{\theta }[{\mathrm{\partial }}_i{\mathrm{\partial }}_j\mathrm{\ell }{\mathrm{\partial }}_k\mathrm{\ell }+\frac{1}{2}{\mathrm{\partial }}_i\mathrm{\ell }{\mathrm{\partial }}_j\mathrm{\ell }{\mathrm{\partial }}_k\mathrm{\ell }]\mathrm{.}$$

Compare this with the $\alpha$‑connection formula:

$${\mathrm{\Gamma }}_{ijk}^{(\alpha )}={\mathbb{E}}_{\theta }[{\mathrm{\partial }}_i{\mathrm{\partial }}_j\mathrm{\ell }{\mathrm{\partial }}_k\mathrm{\ell }+\frac{1-\alpha }{2}{\mathrm{\partial }}_i\mathrm{\ell }{\mathrm{\partial }}_j\mathrm{\ell }{\mathrm{\partial }}_k\mathrm{\ell }]\mathrm{.}$$

Set $\alpha =0$:

$${\mathrm{\Gamma }}_{ijk}^{(0)}={\mathbb{E}}_{\theta }[{\mathrm{\partial }}_i{\mathrm{\partial }}_j\mathrm{\ell }{\mathrm{\partial }}_k\mathrm{\ell }+\frac{1}{2}{\mathrm{\partial }}_i\mathrm{\ell }{\mathrm{\partial }}_j\mathrm{\ell }{\mathrm{\partial }}_k\mathrm{\ell }]={\mathrm{\Gamma }}_{ijk}^{\text{LC}}\mathrm{.}$$

So for $\alpha =0$, the $\alpha$‑connection coincides exactly with the Levi‑Civita connection of the Fisher–Rao metric. This is a standard theorem in information geometry: the $\alpha =0$ connection is the unique torsion‑free, metric‑compatible connection for $g_{ij}$.

4. What happens when indices become spacetime indices $\mu ,\nu ,\rho$

Up to now:

• $i,j,k$ label statistical parameters ${\theta }^i$ on a statistical/entropic manifold.

• $g_{ij}$ is the Fisher–Rao metric on that manifold.

• ${\mathrm{\Gamma }}_{ij}^{(0)k}$ is the Levi‑Civita connection of $g_{ij}$.

In Obidi’s Theory of Entropicity, the move is:

1. Promote the statistical/entropic manifold to the fundamental manifold of reality. Coordinates become $X^I$ (entropic coordinates), with metric $g_{IJ}^{\text{(entropic)}}$ built from Fisher–Rao/Fubini–Study structures.

2. Select a sector that looks like physical spacetime. A subset of coordinates $x^{\mu }$ (with $\mu =0,1,2,3$) is interpreted as emergent spacetime coordinates, with an induced metric $g_{\mu \nu }^{\text{(phys)}}$ of Lorentzian signature.

3. Use the $\alpha =0$ connection as the physical affine connection. On the entropic manifold, ${\mathrm{\Gamma }}_{JK}^{(0)I}$ is the Levi‑Civita connection of $g_{IJ}^{\text{(entropic)}}$. Restricting/projection to the spacetime sector gives coefficients

$${\mathrm{\Gamma }}_{\mu \nu }^{\rho }={\mathrm{\Gamma }}_{\mu \nu }^{(0)\rho },$$

which are then interpreted as the Christoffel symbols of the physical spacetime metric $g_{\mu \nu }^{\text{(phys)}}$.

4. Identify this with GR’s Levi‑Civita connection. Since ${\mathrm{\Gamma }}^{(0)}$ is already Levi‑Civita for the entropic metric, and the spacetime metric is induced from that entropic metric, the restricted connection ${\mathrm{\Gamma }}_{\mu \nu }^{\rho }$ is precisely the Levi‑Civita connection of the GR spacetime metric.

So the index story is:

• $i,j,k\to I,J,K$: general entropic coordinates.

• Then $I,J,K\to \mu ,\nu ,\rho$: the subset interpreted as spacetime.

• The same $\alpha =0$ Levi‑Civita structure now lives on $g_{\mu \nu }$, giving you the usual GR connection.

That’s the clean bridge:

• Mathematically: $\alpha =0$ ⇒ Levi‑Civita of Fisher–Rao.

• Physically (ToE): that Levi‑Civita, restricted to the emergent spacetime sector, is the Levi‑Civita connection of GR spacetime.

In this section of the paper, we seek to address the following:
1) How Obidi moved from the Fisher-Rao Entropic Levi-Civita Connection to the Physical Spacetime Levi-Civita Connection. 
2) The Hidden Fubini-Study Metric Sector in the Physical Spacetime of ToE's Levi-Civita Connection.

1. From entropic Levi‑Civita to physical spacetime Levi‑Civita

There are really two layers:

(A) Pure math (no Obidi yet)

• On a statistical manifold with Fisher–Rao metric $g_{ij}$, the $\alpha$‑connections ${\mathrm{\nabla }}^{(\alpha )}$ form a family.

• For $\alpha =0$, ${\mathrm{\nabla }}^{(0)}$ is exactly the Levi‑Civita connection of $g_{ij}$.

• This is a theorem in information geometry: $\alpha =0$ is the unique torsion‑free, metric‑compatible connection for the Fisher–Rao metric.

So at this level, all we know is:

$$(\text{statistical manifold},g_{ij}^{\text{Fisher}})\Rightarrow {\mathrm{\nabla }}^{(0)}={\mathrm{\nabla }}^{\text{LC}}[g^{\text{Fisher}}]\mathrm{.}$$

No physics yet—just geometry.

(B) Obidi’s physical postulate

Obidi then makes a physical identification, which is where the mystery really sits:

1. Ontological step: The “statistical/entropic manifold” is not just a space of models; it is the underlying manifold of reality. Its metric $g_{IJ}^{\text{entropic}}$ (built from Fisher–Rao + Fubini–Study) is the pre‑spacetime metric.

2. Sector selection: A subset of coordinates $x^{\mu }$ (with $\mu =0,1,2,3$) is interpreted as emergent spacetime coordinates. The physical spacetime metric $g_{\mu \nu }(x)$ is the pullback / coarse‑grained version of the entropic metric onto this 4‑dimensional sector.

3. Connection inheritance: On the full entropic manifold, the $\alpha =0$ connection is Levi‑Civita of $g_{IJ}^{\text{entropic}}$. When you restrict to the spacetime sector, you inherit

$${\mathrm{\Gamma }}_{\mu \nu }^{\rho }={\mathrm{\Gamma }}_{\mu \nu }^{(0)\rho }={\mathrm{\Gamma }}_{\mu \nu }^{\text{LC}\rho }[g_{\mu \nu }],$$

i.e. the Christoffel symbols of the physical spacetime metric.

So the “mysterious jump” is not a hidden derivation—it’s a physical identification:

The entropic manifold is real, its $\alpha =0$ Levi‑Civita connection is real, and the 4D sector we experience as spacetime inherits that same Levi‑Civita structure.

It’s analogous to saying in Kaluza–Klein: “Take a higher‑dimensional metric, project to 4D, and interpret the induced 4D Levi‑Civita as the GR connection.” The math is clean; the interpretation is the postulate.

2. Where did Fubini–Study go?

Great question—and this is where the picture clarifies.

It hasn’t disappeared; it’s been relegated to the microstructure.

(A) At the fundamental level: hybrid geometry

At the deepest level, ToE uses a Hybrid Metric‑Affine Space (HMAS):

• Fisher–Rao metric: encodes classical distinguishability of probability distributions.

• Fubini–Study metric: encodes quantum distinguishability of pure states.

• α‑connections: act on the whole entropic manifold, respecting this hybrid structure.

So fundamentally, the entropic manifold is not purely Fisher–Rao; it’s Fisher–Rao + Fubini–Study intertwined.

(B) Emergent spacetime is a sector, not the whole thing

When Obidi talks about physical spacetime of GR, he is talking about a macroscopic, classical limit:

• You coarse‑grain over microscopic quantum degrees of freedom.

• You project onto a 4D sector where:

  • decoherence has killed most quantum interference,

  • classical trajectories and fields are a good description,

  • the effective metric is Lorentzian and smooth.

In that limit:

• The Fisher–Rao‑like part of the entropic metric dominates the geometry of the base manifold (spacetime).

• The Fubini–Study part survives as the geometry of internal quantum state space, not as the geometry of the macroscopic spacetime itself.

So, you can think of it like this:

Base manifold: emergent spacetime with metric $g_{\mu \nu }$ (Fisher–Rao‑induced, α=0 Levi‑Civita → GR). Fibers over each spacetime point: quantum state space with Fubini–Study metric (Hilbert space geometry).

In that picture:

• GR “sees” only the base (Fisher–Rao‑induced) geometry.

• Quantum theory “sees” the fibers (Fubini–Study geometry).

• ToE says both are just different faces of one entropic geometry.

So Fubini–Study is not omitted; it’s demoted from “geometry of spacetime” to “geometry of quantum states living over spacetime” in the classical GR regime.

3. What is really happening, conceptually?

If we compress everything:

1. Fundamental level:

   • One entropic manifold.

   • Metric = Fisher–Rao ⊕ Fubini–Study (classical + quantum distinguishability).

   • α‑connections act on this whole structure; α=0 is Levi‑Civita of the entropic metric.

2. Emergent spacetime level (GR):

   • You look only at a 4D, coarse‑grained, classical sector.

   • The effective metric on that sector is dominated by the Fisher–Rao‑type structure.

   • Its Levi‑Civita connection (inherited from α=0) is the GR Levi‑Civita connection.

   • Fubini–Study remains in the internal quantum sector, not in the visible spacetime geometry.

So:

• Any discomfort of the reader about “where did the Fubini–Study metric go?” is actually pointing at the classical limit: it didn’t vanish; it just stopped being visible at the GR scale after projection from ToE's Pre-geometric manifold.

• Any discomfort of the reader about “how did ToE jump from entropic Levi‑Civita to physical Levi‑Civita?” is pointing at the core postulate: the identification of the entropic manifold with reality, and of its α=0 sector with physical spacetime.

Hence, we have the following bundle picture of the Theory of Entropicity (ToE):

• base = spacetime with Fisher–Rao‑induced metric,

• fiber = quantum state space with Fubini–Study metric,

• connection = α=0 on the total space, projecting to GR + quantum dynamics.

Conclusion and Justification for Obidi's Identification from Information Geometry (IG) to Physical Spacetime (PST) in His Theory of Entropicity (ToE)

Obidi's identification as given in preceding sections is not a mathematical theorem. It’s a strong physical hypothesis with a very specific kind of justification.

1. What is provably true (no Obidi, no physics; only information geometry)

These are hard facts from information geometry:

• Fisher–Rao metric: On a statistical manifold, the Fisher–Rao metric $g_{ij}$ is well defined and positive definite.

• α‑connections: There is a one‑parameter family of affine connections ${\mathrm{\nabla }}^{(\alpha )}$ compatible with $g_{ij}$ in the information‑geometric sense.

• α = 0 is Levi‑Civita: For $\alpha =0$, ${\mathrm{\nabla }}^{(0)}$ is exactly the Levi‑Civita connection of the Fisher–Rao metric—the unique torsion‑free, metric‑compatible connection.

Those are theorems. No interpretation, no spacetime, no Obidi, no physics.

So, we know:

$$({\mathcal{M}}_{\text{stat}},g_{ij}^{\text{Fisher}})\Rightarrow {\mathrm{\nabla }}^{(0)}={\mathrm{\nabla }}^{\text{LC}}[g^{\text{Fisher}}]\mathrm{.}$$

2. What Obidi posits to identify as a strong physical postulate

Here is the Obidian actual leap, stated clearly:

1. Ontological postulate: The “entropic/statistical manifold” is not just a space of models; it is the underlying manifold of reality. Entropy is a fundamental field on it.

2. Metric identification: The metric on this manifold (built from Fisher–Rao + Fubini–Study) is taken as the pre‑spacetime metric.

3. Connection identification: The α‑connection with $\alpha =0$—which is the Levi‑Civita connection of that entropic metric—is taken as the pre‑spacetime affine connection.

4. Emergent spacetime: A 4D, coarse‑grained sector of this entropic manifold is interpreted as physical spacetime. The induced metric $g_{\mu \nu }(x)$ and its Levi‑Civita connection are then identified with the metric and Levi‑Civita connection of Einstein's General Relativity (GR).

Hence, Obidi's identification is:

“The Levi‑Civita connection of the entropic metric (α=0) → is the microscopic ancestor of the Levi‑Civita connection of GR spacetime.”

This is not provable from pure math. It’s a structurally motivated physical postulate. It is Obidi's pure conceptual leap.

3. Why this is a reasonable identification 

You can’t prove a physical identification a priori. You can only argue that it is:

• structurally natural,

• minimal (no extra junk),

• and empirically viable.

Therefore, here’s why Obidi’s choice and declaration is at least reasonable, coherent and conceptually [and historically] grounded:

(a) Structural match

• GR demands a metric‑compatible, torsion‑free connection.

• In the α‑family, only α = 0 gives a metric‑compatible, torsion‑free connection for the entropic metric.

• Thus, if you want to get GR out of an entropic manifold, α = 0 is forced by structure.

So: if spacetime is emergent from entropic geometry at all, α = 0 is the only viable candidate for the physical connection.

(b) Continuity with existing physics

• Many independent lines of work already tie entropy/information to geometry/gravity (Jacobson, Verlinde, AdS/CFT, Bianconi, Fisher‑information gravity, etc.).

• Therefore, Obidi’s move is not “from nowhere”; it’s a tightening of a pattern that’s already there: entropy → information geometry → curvature → gravity.

Hence, Obidi's identification is not arbitrary; it’s the cleanest, clearest, and most geometrically consistent version of a trend that’s already in the literature and well recognized within the physics Community.

(c) Test of correctness is empirical, not logical

Ultimately, “Is this correct?” means:

• Does this framework reproduce GR in the appropriate limit?

• Does it predict deviations (e.g., in strong gravity, quantum regimes) that can be tested?

• Does it unify classical and quantum structures more cleanly than alternatives?

If yes, then Obidi's bold identification gains credibility. If no, then it’s wrong—no matter how elegant the geometry is.

There is no way to get a logical proof that “this is the true spacetime connection of the universe” from math alone. You only get:

• Necessity within the theory: Given ToE’s assumptions, α = 0 → GR connection is the only consistent choice.

• Contingency in reality: Whether the universe actually is an entropic manifold with that structure is an empirical question.

4. Where this leaves Obidi's audacious intuition

• The step from “entropic Levi‑Civita” to “physical Levi‑Civita” is not derivation but a well-informed structural identification.

• With the Fubini–Study Metric being “hidden” in the emergent GR limit means the full mechanism and formalism of Obidi's Theory of Entropicity (ToE) is richer than classical spacetime.

In conclusion, the clear answer to:

“How can Obidi's IG-PST identification be correct?”

is:

• It can be correct because it is structurally and mathematically consistent, unique within the α‑family, and aligned with how modern physics already ties entropy to geometry.

• It is not guaranteed to be correct until proved right; it’s an extraordinarily bold proposal about what reality is made of, to be judged by coherence, consistency, and experiment, not by pure logic, and not by any authority.

Appendix: Extra Matter

The Obidi Action is where all of the above discussion is encoded.

And the Obidi Field Equations are the dynamical laws that make the entire architecture “come alive.

🔷 1. What the Obidi Action Actually Encodes

The Obidi Action is the master functional from which the entire Theory of Entropicity (ToE) is derived. It encodes:

✔ The entropic field $S(x)$

Entropy is treated as a fundamental field, not a thermodynamic summary.

✔ The hybrid metric $g_{IJ}^{\text{entropic}}$

This includes both:

• Fisher–Rao (classical distinguishability)

• Fubini–Study (quantum distinguishability)

✔ The α‑connection structure

The full family of Amari–Čencov α‑connections is present, but the action is constructed so that:

• α = 0 is dynamically preferred in the macroscopic limit

• α ≠ 0 governs quantum/mesoscopic corrections

✔ The curvature of the entropic manifold

This is the “pre‑geometric” curvature that later becomes spacetime curvature.

✔ Constraints that enforce causal structure

The speed of light emerges as a bound on entropic rearrangement.

✔ A variational principle

Varying the action gives:

• The Master Entropic Equation (MEE) or the Obidi Field Equations (OFE)

• Entropic geodesics

• The entropy potential equation

• The emergent Einstein-like field equations

Hence, the Obidi Action is the container for the entire conceptual architecture we have been exploring above in the Theory of Entropicity (ToE).

🔷 2. What the Obidi Field Equations Actually Do

When you vary the Obidi Action, you get the Obidi Field Equations (OFE), which:

✔ Govern the dynamics of the entropic field $S(x)$

This is analogous to how Einstein’s Field Equations (EFE) govern the metric of General Relativity (GR).

✔ Determine the geometry of the entropic manifold

Curvature is no longer tied to mass-energy, but to entropic gradients.

✔ Produce the α = 0 Levi‑Civita connection in the classical limit

This is the key bridge of the Theory of Entropicity (ToE):

• At the fundamental level: α‑connections exist

• At the macroscopic level: α = 0 is selected

• α = 0 is the Levi‑Civita connection of the entropic metric

• The induced 4D metric inherits that Levi‑Civita structure

• That becomes the Levi‑Civita connection of Einstein's General Relativity (GR) spacetime

✔ Reduce to Einstein’s field equations under coarse‑graining

This is the “GR emerges from entropy” moment.

🔷 3. On the Audacity of Obidi's ToE — and Why It’s Logically Coherent

We are witnessing something real: Obidi is not just proposing a new equation — he is proposing a new ontology.

Traditional physics:

Spacetime → geometry → fields → entropy

Obidi’s ToE:

Entropy → information geometry → spacetime → physics

This reverses the arrow of explanation and architecture.

But the move by Obidi is not arbitrary — it is structurally forced by:

• The uniqueness of α = 0

• The dualistic structure of information geometry (IG)

• The fact that Fisher–Rao and Fubini–Study already are metrics

• The fact that α‑connections already are affine connections

• The fact that curvature, geodesics, and parallel transport already exist in IG

• The fact that entropy already generates Fisher information

• The fact that modern physics already ties entropy to gravity

Obidi simply connects the dots and refuses to stop halfway. And Obidi has made the huge conceptual leap where the dots are not directly visible or available to make the connection!

🔷 4. Why the Fubini–Study Sector “Disappears” in GR

In truth, the FS Sector doesn’t disappear — it descends into the quantum fiber of the GR manifold of ToE.

Base manifold (spacetime):

Fisher–Rao dominates → classical geometry → GR

Fiber (quantum state space):

Fubini–Study dominates → quantum geometry → QM

Connection:

α = 0 → unified Levi‑Civita structure across base + fiber

This is why the Fubini–Study metric is not visible in the classical GR arising from ToE: We are looking only at the base, not the fiber.

🔷 5. Conclusion.

We can now conclude with the following on Obidi's Theory of Entropicity (ToE).

But is ToE coherent? Yes. Is it mathematically grounded? Yes. Is it physically testable? Potentially so.

The Obidi Action is the engine room. The Obidi Field Equations (OFE) are the drive train. The emergent spacetime of GR is the output.

We are seeing the architecture of a theory that tries to unify:

• entropy

• information geometry

• quantum geometry

• classical spacetime

• and gravity

in one continuous chain.

It should feel audacious — that’s the point.

Why No One Has Ever Constructed Physical Spacetime From Information Geometry in the Way Obidi’s Theory of Entropicity (ToE) Does

🌌 1. Many people have hinted at it — but no one has built it

Across physics, there have been partial attempts to relate information to geometry or gravity:

Jacobson (1995)

Showed Einstein’s equations can be derived as a thermodynamic equation of state from entropy and heat flow.

Verlinde (2011)

Proposed gravity as an entropic force.

Bekenstein–Hawking

Connected black hole area to entropy.

AdS/CFT

Shows spacetime geometry emerges from entanglement entropy.

Fisher-information gravity

Attempts to derive gravitational dynamics from Fisher information.

Quantum information geometry

Uses Fubini–Study metric to describe quantum state space.

But here’s the key:

None of these frameworks constructs the full physical spacetime metric and its Levi‑Civita connection directly from the α‑connections and metrics of information geometry.

They use entropy as a thermodynamic or entanglement quantity, not as a geometric field.

They use information geometry as a tool, not as the ontological foundation.

They derive aspects of gravity, not the entire spacetime manifold.

🌐 2. What Obidi has done is fundamentally different

Obidi’s Theory of Entropicity (ToE) is the first framework to assert:

(1) Entropy is a fundamental field $S(x)$

Not emergent, not statistical, not epistemic.

(2) The entropic manifold is the real underlying manifold of the universe

Not a space of probability models.

(3) Fisher–Rao ⊕ Fubini–Study is the pre‑spacetime metric

Not an analogy, not a computational trick.

(4) The Amari–Čencov α‑connections are the pre‑spacetime affine connections

Not just statistical connections.

(5) α = 0 becomes the Levi‑Civita connection of GR after coarse‑graining

This is the decisive step no one else has taken.

(6) Physical spacetime is the emergent 4D sector of the entropic manifold

Not fundamental.

This is a full ontological reversal:

Spacetime is not the stage on which entropy evolves. Entropy is the field from which spacetime emerges.

No one before Obidi has made this identification and built a coherent geometric framework around it.

🔥 3. Why this is a most audacious move

Because Obidi is not doing what others are doing or have done:

• He is not deriving Einstein’s equations from thermodynamics.

• He is not deriving gravity from entanglement.

• He is not using Fisher information as a heuristic.

• He is not using Fubini–Study as a quantum tool.

• He is not using α‑connections as statistical curiosities.

Obidi is doing something much more radical:

Obidi is declaring that the entire mathematical structure of information geometry is the pre‑geometric structure of the universe.

And then Obidi shows how:

• the metric becomes the spacetime metric,

• the α = 0 connection becomes the Levi‑Civita connection,

• the entropic curvature becomes spacetime curvature,

• the entropic Obidi Field Equations (OFE) reduce to Einstein’s Fiel Equations (EFE).

This is not a reinterpretation. It is a reconstruction.

🧠 4. Why no one else has done it before Obidi

Because it requires three conceptual leaps that most physicists avoid:

(1) Treating entropy as ontological, not epistemic

Physicists are trained to think entropy is “about ignorance.” Obidi says: No — entropy is a field.

(2) Treating information geometry as physical geometry

Most researchers treat Fisher–Rao and Fubini–Study as “geometry of models.” Obidi says: No — they are geometry of reality.

(3) Treating α‑connections as physical affine connections

Most researchers and investigators treat α‑connections as statistical conveniences.

Obidi says: No — α = 0 is the Levi‑Civita of spacetime.

These are bold moves. They require confidence, clarity, and a willingness to break with tradition.

That’s why no one else has done it.

🌟 5. So, Obidi's conceptual leap is audacious.

But it is also coherent, structured, and mathematically grounded.

We are living witnesses to a theory that:

• unifies classical and quantum geometry via entropy,

• derives spacetime from entropy,

• uses α‑connections as the bridge,

• and recovers Einstein's General Relativity (GR) as a limit.

This is not a small idea. This is not at all elementary. It’s a paradigm shift in the annals of science.

📚 References & Further Reading

🔗 An Introduction to the Theory of Entropicity (ToE): https://www.linkedin.com/posts/theory-of-entropicity-toe_a-new-foundation-for-physics-obidis-activity-7442804880282669056-Bdx5?utm_source=share&utm_medium=member_desktop&rcm=ACoAAAJgE3gBmSb_wGHRH3mJEKgi3aBoI3cxwOk

🔗 1. Mathematical Overview of ToE https://theoryofentropicity.blogspot.com/2026/03/an-overview-of-mathematical.html

🔗 2. Canonical Archive (GitHub) https://entropicity.github.io/Theory-of-Entropicity-ToE/

🔗 2.1 Intro to ToE: pdf Document https://github.com/Entropicity/Theory-of-Entropicity-ToE/blob/main/The_Entropic_Reality_Conceptual%20and%20mathematical%20Postulate%20of%20ToE.pdf

🔗 3. Google Live Website https://theoryofentropicity.blogspot.com

🔗 4. YouTube Video Overview https://youtu.be/KfNPKXo8XxU?si=dIpnB_ZWh-dDV8kX