⭐ The Obidi Equivalence Principle (OEP) of the Theory of Entropicity (ToE)

Spacetime is the macroscopic projection of an underlying information‑geometric manifold, and every geometric property of physical spacetime corresponds to an entropic property of that information manifold.

More formally:

> The curvature, geodesics, and metric structure of physical spacetime arise from, and are isomorphic to, the curvature, geodesics, and Fisher‑information metric of the underlying information‑geometric manifold after coarse‑graining.

The Obidi Equivalence Principle proposes a global isomorphism between an underlying information-geometric manifold endowed with a Fisher metric and emergent physical spacetime, asserting that all geometric and dynamical properties of spacetime correspond to entropic and informational structures under a coarse-graining map.

🧠 Collective Historical Insight Summary

What ToE has formulated as the Obidi Equivalence Principle (OEP) is essentially an attempt to do for entropy/information what Einstein did for gravity:

Establish a strict equivalence (isomorphism) between two domains:

1) information geometry

2) physical spacetime geometry

This idea does have precedents in fragments across physics:

holography (geometry ↔ entanglement)

AdS/CFT (bulk ↔ boundary)

information geometry (Fisher metric ↔ statistical structure)

But:

👉 No mainstream framework currently enforces a full, global, invertible isomorphism of the kind which Obidi's Theory of Entropicity (ToE) has demanded.

So ToE's OEP is:

not baseless

but much stronger than anything currently accepted in traditional physics.

✅ Legitimacy of the Use of the Fisher Information Metric in the Theory of Entropicity (ToE)

From established literature:

Amari (2016) shows Fisher information defines a Riemannian metric

Anza & Crutchfield (2022) connect entropy and geometric structure

Franzosi et al. (2016) define geometric entropy via curvature

👉 So ToE's starting point:

“information → geometry”

is fully grounded in existing research

✅ ToE is Attempting a True Equivalence Principle

This is both important and audacious at once.

Compare:

Einstein EP

ToE's OEP

gravity ≡ geometry (Einstein)

spacetime ≡ information geometry (Obidi)

inertial = gravitational mass  (Einstein)

physical = entropic geometry (Obidi)

local equivalence  (Einstein)

global mapping (Obidi)

👉 Structurally, this is the right kind of move for a foundational theory.

This is the core demand of the Theory of Entropicity (ToE):

\[

(\mathcal{M}{info}, g{F}) \;\xrightarrow{\text{emergence}}\; (\mathcal{M}{spacetime}, g{\mu\nu})

\]

with the requirement that:

\[

\Phi: \mathcal{M}{info} \to \mathcal{M}{spacetime}

\]

is a smooth, invertible mapping preserving curvature, geodesics, and entropy production.

That is:

Emergence relation:

(𝓜₍ᵢₙfₒ₎, gF) ⟶₍ₑₘₑᵣgₑₙcₑ₎ (𝓜₍ₛₚₐcₑₜᵢₘₑ₎, gμν)

Correspondence map:

Φ : 𝓜₍ᵢₙfₒ₎ → 𝓜₍ₛₚₐcₑₜᵢₘₑ₎

⭐ What the principle asserts (in plain language)

1. Information geometry is the substrate.  

   It exists before spacetime.

2. Spacetime emerges from information geometry.  

   Not as a metaphor — as a coarse‑grained projection.

3. Every physical geometric quantity has an entropic counterpart.  

   - Spacetime curvature ↔ Entropic curvature  

   - Geodesics ↔ Paths of extremal entropy flow  

   - Mass ↔ Information‑curvature density  

   - Energy ↔ Rate of information change  

4. Gravity is not a force but an entropic gradient.  

   This is the entropic analogue of Einstein’s “gravity = geometry.”

5. The arrow of time is the monotonic increase of Fisher information.

⭐ Why this principle is necessary for the Theory of Entropicity (ToE)

Without the OEP:

- spacetime and information geometry become dualistic  

- entropy cannot be the fundamental invariant  

- gravity cannot be entropic  

- quantum mechanics cannot be geometric  

- the ToE collapses into two incompatible layers  

With the Obidi Equivalence Principle (OEP):

- quantum → statistical geometry  

- gravity → entropic curvature  

- time → information ordering  

- energy → information flow  

- spacetime → emergent macro‑geometry  

Everything becomes one coherent structure.

⭐ The 3 Axioms of the Theory of Entropicity (ToE):

- Axiom 1: Entropic Primacy  

- Axiom 2: Information‑Geometric Substrate  

- Axiom 3: Obidi Equivalence Principle (OEP)  

References 

1)

https://theoryofentropicity.blogspot.com/2026/04/obidi-equivalence-principle-oep.html

2)

http://youtube.com/post/UgkxMndtJGNXut5AISg1-2nRtwCBDUmhkxYM?si=xnYZNWea-SgUSb9M

3)

https://medium.com/@jonimisiobidi/the-obidi-equivalence-principle-oep-of-the-theory-of-entropicity-toe-8ff4c199a3d7