🚀 A Demonstration of the Theory of Entropicity (ToE)'s Core Claim that Spacetime and Its Curvature Cannot Exist Without Underlying Entropic Gradients: Spacetime as an Entropic Phenomenon

In information geometry and John Onimisi Obidi’s ToE, spacetime curvature is not treated as an inherent property of empty space. Instead, curvature is constructed from the algebraic divergence between two dual statistical connections — the Amari–Čencov +1 (mixture) and −1 (exponential) connections — acting on an underlying entropic statistical manifold. This dual‑connection structure shows that curvature is a derived quantity, emerging only when informational updates collide.

This is the heart of ToE’s claim:  

No entropic gradients→No Čencov tensor → No curvature→No gravity.

🔶 1️⃣ Dual Connections and the Čencov Structural Tensor

On a statistical manifold with Fisher information metric gᵢⱼ, the Amari–Čencov connections are:

- Mixture connection (+1):  

  Γ⁽¹⁾ = Γ⁽⁰⁾ + ½·C

- Exponential connection (−1):  

  Γ⁽⁻¹⁾ = Γ⁽⁰⁾ − ½·C

Here, Cᵢⱼₖ is the Čencov structural tensor — the “entropic curvature generator.” These two connections represent opposing informational update geometries: one linear (mixture), one exponential (log‑linear). Their divergence encodes the entropic structure of the manifold.

🔶 2️⃣ The Riemann Curvature Tensor

The curvature of any affine connection is defined by the failure of covariant derivatives to commute:

R = ∂Γ + Γ·Γ − (terms with k ↔ l)

If either the +1 or −1 connection is individually flat (as in exponential families), its curvature vanishes. Yet physical curvature does not vanish — meaning it must arise from the interaction between the two dual connections. Key insight: curvature is not a primitive geometric axiom but a statistical consequence of entropic asymmetry.

🔶 3️⃣ The Explicit Construction: Curvature = Clash of Dual Structures

Obidi shows that the physical Riemann curvature tensor is:

R⁽⁰⁾ = ½·(R⁽¹⁾ + R⁽⁻¹⁾) − ¼·(C·C − C·C)

In ToE’s informationally flat substrate:

- R⁽¹⁾ = 0  

- R⁽⁻¹⁾ = 0

So the physical curvature reduces to:

R⁽⁰⁾ = −¼·(C × C)

Thus:

> Spacetime curvature is literally the antisymmetrized product of Čencov tensors — the “friction” between mixture and exponential information geometries.

The tensor Cᵢⱼₖ acts as the entropic “shear” that generates curvature when informational flows disagree.

🔶 4️⃣ The Physical Interpretation

🔹 Gravity = Entropic Friction

Curvature emerges from the algebraic clash between the +1 and −1 informational update rules. This “friction” is encoded in the Čencov tensor. Gravity is therefore the macroscopic geometric shadow of microscopic entropic divergence.

🔹 No Entropy→No Curvature

If the entropic field is uniform:

- Cᵢⱼₖ = 0  

- ⇒ Rᵢⱼₖₗ = 0  

- ⇒ spacetime becomes flat (Minkowski)

This is a direct mathematical demonstration of ToE’s core claim:

> Spacetime curvature cannot exist without underlying entropic gradients.

Gravity is not a primitive force—it is emergent from entropy.

For Details:
📚Reference(s):
The Canonical Archives: https://entropicity.github.io/Theory-of-Entropicity-ToE/