On the Mathematical Foundation of the Theory of Entropicity (ToE): The Historical Reflection, Logical Motivation, and Conceptual Leap in Relation to Contemporary Researchers and Investigators 

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🧠 Collective Insight Summary

The researchers mentioned below:

• Amari (2016)
• Anza & Crutchfield (2022)
• Franzosi et al. (2016)

all establish an important fact:

Entropy and information naturally generate geometric structures.

But they stop at geometry as a mathematical or descriptive tool.

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👉 What  ToE (Obidi)  has done is fundamentally different:

Obidi is not using information geometry to describe systems —
Obidi is asserting that information geometry is physical reality itself.

That is the key distinction.

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🔍 What Existing Researchers Have Actually Done 

1. Amari (2016) — Information Geometry

• Shows:
  • Fisher information defines a Riemannian metric
  • Probability distributions form statistical manifolds

👉 But:

• This geometry lives in parameter space, not spacetime
• It is a tool for inference and statistics

📌 It does NOT claim:

spacetime = Fisher geometry

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2. Anza & Crutchfield (2022) — Entropy & Geometry

• Connect:
  • entropy
  • information dimension
  • geometric structure

👉 But:

• Focus is on quantum systems and complexity
• Geometry is derived from informational properties

📌 Again:

geometry is descriptive, not ontological

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3. Franzosi et al. (2016) — Geometric Entropy

• Define:
  • entropy measures based on curvature of manifolds

👉 But:

• Applied to:
  • networks
  • complex systems

📌 Geometry is:

a way to measure complexity, not the fabric of spacetime

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🔥 What Obidi Has Done That Is Different

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⭐ 1. Obidi Made an Ontological Leap

Others:

Information geometry describes systems

Obidi:

Information geometry is the substrate of reality

That is a category shift, not just an extension.

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⭐ 2. Obidi Imposed a Full Equivalence [the Obidi Equivalence Principle (OEP)]

Others:

• Explore relationships:
  • entropy ↔ geometry
  • information ↔ structure

Obidi:

Demands a one-to-one correspondence (isomorphism):

(\mathcal{M}_{info}, g_F) \leftrightarrow (\mathcal{M}_{spacetime}, g_{\mu\nu})

👉 No one in mainstream literature enforces this as a strict principle.

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⭐ 3. Obidi Promotes Geometry → Physics

Others:

• Geometry = mathematical structure

Obidi:

• Geometry = physical dynamics

Specifically:

• geodesics → physical motion
• curvature → gravity
• entropy flow → time evolution

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⭐ 4. Obidi Closes the Loop with an Action Principle

Others:

• Do not generally provide:
  • a full field theory based on entropy geometry

Obidi:

• Introduce:
  • entropic action
  • field equations (MEE)

👉 This attempts to turn:

information geometry → dynamical physics

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⭐ 5. Obidi Eliminates Dualism

Most research still has:

Layer	Role
Physical spacetime	real
Information geometry	descriptive

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Obidi replaces this with:

Single Layer
Information geometry = spacetime

👉 That’s a monistic framework, not dual.

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⚖️ Statement of the Difference

Here, we state the essential difference between what Obidi has done and what other researchers and investigators have done:

Existing research shows that entropy and information can be represented geometrically. 

The Theory of Entropicity (ToE) goes further by asserting that this information geometry is not merely representational but is physically real, and that spacetime itself is an emergent, isomorphic projection of this underlying entropic manifold governed by its own action and field equations.

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Obidi's Challenge and Responsibility 

This distinction made by Obidi is indeed real—but it comes with audacious responsibility and ontological courage.

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1. Obidi Made a Stronger Claim Than Others

They say:

• “geometry models information”

Obidi says:

• “geometry is reality”

👉 That requires:

• stricter proof
• stronger constraints

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2. Obidi Introduced Isomorphism (Very Strong)

Most researchers avoid claiming:

• invertible mapping
• full preservation of structure

👉 Because this is extremely hard to justify mathematically.

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3. Obidi Enters the “Theory of Everything” Zone

By unifying:

• spacetime
• entropy
• information

👉 Obidi's ToE is competing with:

• GR
• QFT
• quantum gravity programs

🧩 The Deep Philosophical Shift Obidi is Proposing

Standard View:

Reality → information

Obidi's View:

Information → reality

Standard View:

Geometry is physical

Obidi's View:

Geometry is informational

Standard View:

Entropy is derived

Obidi's View:

Entropy is fundamental

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📚 Key Sources

1. Amari, S. (2016). Information Geometry and Its Applications. Springer.
2. Anza, F., & Crutchfield, J. P. (2022). Quantum information dimension and geometric entropy. PRX Quantum.
3. Franzosi, R. et al. (2016). Riemannian geometric entropy. Phys. Rev. E.
4. Felice, D. et al. (2018). Information geometric methods for complexity. Chaos.
5. Axelkrans, E. (2025). Emergent spacetime as information geometry. PhilArchive.
6. Arneth, B. (2025). Entropy, topology, and origins of geometry. HAL.
7. Ashkenazy, L. (2024). Drive-field information theory.
8. Lesne, A. (2014). Shannon entropy… MSCS.
9. Carroll, S. & Remmen (2016). What is entropy in entropic gravity?
10. Obidi, J. O. (2025). Foundations of ToE.

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🧠 Conclusion 

👉 Obidi did not just extend information geometry
👉 Obidi reinterpreted it as physical reality itself

That is the real difference and audacity of Obidi's Theory of Entropicity (ToE).

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