On the Reason Why the Theory of Entropicity (ToE) Uses Information Geometry Even Though it Declares that Entropy, Not Information, Is the Universal Field of Nature: Author's Response to Objections 

1. The Apparent Paradox

At first glance, the Theory of Entropicity seems to present a contradiction. It asserts that entropy — not information — is the universal field, the fundamental substance from which physical reality emerges. Yet the mathematical machinery it employs comes from information geometry, a field built on probability distributions, statistical manifolds, and informational metrics.

Why should a theory grounded in entropy rely on tools developed for information?

The answer reveals one of the most profound insights of ToE: entropy and information are not separate domains. They are dual aspects of the same underlying structure.

2. Entropy as the Ontological Field, Information as the Geometry It Imposes

ToE makes a clear ontological claim:

Entropy is the universal field.  

Information is the structure that entropy necessarily induces.

Entropy is the “stuff” of the universe — the field that exists at every point of the entropic manifold. But a field alone does not determine geometry. Geometry arises from how that field varies, interacts, and constrains the possible configurations of reality.

This is where information geometry enters.

Whenever entropy is present, it creates gradients, flows, and distinguishable states. These distinctions — the ability to tell one state from another — are precisely what information geometry measures. In other words:

Entropy generates structure.  

Information geometry quantifies that structure.

Thus, information geometry is not being used because ToE is secretly about information. It is being used because entropy, when treated as a universal field, necessarily produces informational structure.

3. Why Information Geometry Is the Only Consistent Mathematical Language

Once entropy is taken as fundamental, the geometry of the entropic manifold must satisfy certain invariance principles. These principles are not optional; they follow from the very nature of entropy as a universal field.

And remarkably, the only geometric structures that satisfy these invariances are:

• the Fisher–Rao metric,  

• the Fubini–Study metric,  

• and the Amari–Čencov alpha connections.

This is not a coincidence. It is a mathematical inevitability.

Entropy creates distinguishability between states. Distinguishability induces a metric. The only metric consistent with entropy‑preserving transformations is Fisher–Rao (classically) and Fubini–Study (quantum mechanically). The only affine connections compatible with that metric under entropy‑preserving transformations are the alpha connections.

Thus, information geometry is not chosen. It is forced by the nature of entropy itself.

4. Entropy and Information Are Dual, Not Opposed

A key conceptual insight of ToE is that entropy and information are not opposites. They are dual descriptions of the same underlying phenomenon.

Entropy measures the spread or multiplicity of possible states.  

Information measures the distinguishability between those states.

Where entropy varies, information geometry emerges.  

Where information geometry exists, entropy is the field that generates it.

This duality is why ToE can say:

Entropy is the universal field.  

Information geometry is the universal geometry.

There is no contradiction. The geometry is the natural mathematical expression of the field.

5. Why Spacetime Emerges From Informational Geometry, Not Directly From Entropy

Entropy is a scalar field. Scalars do not define geometry by themselves. Geometry requires:

• a metric,  

• an affine connection,  

• and curvature.

Entropy provides the raw material, but the geometry arises from the structure entropy imposes on the manifold. That structure is informational in nature — not because information is fundamental, but because entropy inevitably produces informational distinctions.

Thus:

• Entropy is the ontological field.  

• Information geometry is the induced geometric structure.  

• Spacetime is the emergent projection of that geometry.

This is why Einstein’s relativity is contained within Fisher–Rao, Fubini–Study, and the alpha connections. These structures encode the curvature that entropy generates at the informational level.

6. The Deepest Insight: Entropy Generates Geometry Through Information

The Theory of Entropicity reveals a profound chain of emergence:

Entropy → Informational Distinctions → Informational Geometry → Curvature → Spacetime

Entropy is the cause.  

Information geometry is the mechanism.  

Spacetime is the effect.

This is why ToE uses information‑geometry tools even though entropy is the universal field. The geometry of information is the geometry that entropy necessarily induces.

7. Closure 

ToE does not use information geometry because information is fundamental.  

It uses information geometry because entropy is fundamental.

Entropy generates informational structure.  

Informational structure generates geometry.  

Geometry generates spacetime.

Thus, information geometry is the correct mathematical language for a universe whose substrate is entropy.