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.

Why the Synthesis of the Theory of Entropicity (ToE) Was Not Discovered Earlier: Why This Integration Eluded Researchers for Over a Century

1. The Blind Spot Created by Disciplinary Silos

The first and most pervasive reason is structural. The mathematical objects at the heart of the Theory of Entropicity—Fisher–Rao, Fubini–Study, and the Amari–Čencov alpha connections—were developed in completely different intellectual ecosystems.

The Fisher–Rao metric grew out of mathematical statistics.  

The Fubini–Study metric emerged from quantum mechanics and differential geometry.  

The alpha connections were born inside information geometry and statistical inference.

Each of these fields evolved with its own vocabulary, its own problems, and its own philosophical assumptions. Researchers became experts in their own domains, but the conceptual bridges between these domains were never built. The idea that these structures might belong to a single unified geometry was simply not visible from within any one discipline.

ToE emerges precisely because it steps outside these silos and asks a question none of the fields were designed to ask: What if information is not a descriptor of reality, but the substrate of reality?

2. The Epistemic Cage: Information as “Knowledge About” Rather Than “Being Itself”

For most of scientific history, information has been treated as epistemic—a measure of uncertainty, a tool for inference, a way to describe what we know about a system. This epistemic framing created a conceptual cage.

If information is only a descriptor, then the geometry of information is only a geometry of descriptions. It cannot be the geometry of the universe.

This assumption was so deeply embedded that it was rarely questioned. Even when researchers noticed that the Fisher–Rao metric or the Fubini–Study metric had uncanny structural similarities to physical metrics, they interpreted these similarities as coincidences or analogies, not as ontological clues.

ToE breaks the cage by making a single decisive move: it treats information as ontological. Once this shift is made, the geometry of information becomes the geometry of reality, and the entire landscape reorganizes.

3. The Historical Weight of Material Ontology

Physics has been dominated for centuries by a material ontology. Matter, fields, and spacetime were assumed to be the fundamental ingredients of the universe. Information was a secondary concept, a bookkeeping device, a way to quantify ignorance or encode signals.

Under this worldview, it would have been unthinkable to propose that:

• information is the substrate,  

• entropy is a field,  

• Fisher–Rao and Fubini–Study are the metric of reality,  

• alpha connections are the affine structure of the universe,  

• spacetime is an emergent projection of informational curvature.

The conceptual leap required to invert the ontology—from matter to information—was simply too large for earlier frameworks to accommodate. It required a philosophical shift as much as a mathematical one.

4. The Missing Unification: Classical vs Quantum Information Geometry

Even within information geometry, the classical and quantum sectors evolved separately. The Fisher–Rao metric lived in the world of probability distributions. The Fubini–Study metric lived in the world of quantum states. The alpha connections lived in the world of statistical inference.

No one unified these structures because no one believed they belonged to the same manifold. They were seen as analogues, not as components of a single geometry.

ToE is the first framework to assert that these are not analogues but manifestations of one underlying entropic geometry. This unification required stepping outside the assumptions of both classical and quantum information theory.

5. The Lack of a Conceptual Mechanism for Emergence

Even if someone had suspected that informational geometry might be fundamental, there was no clear mechanism for how spacetime could emerge from it. The idea that curvature in an informational manifold could project as gravitational curvature in physical spacetime required a new conceptual toolset: emergence, coarse‑graining, and informational dynamics.

These ideas only matured in the late twentieth and early twenty‑first centuries, through developments in:

• renormalization group theory,  

• holography and emergent gravity,  

• quantum information theory,  

• complexity theory,  

• and the physics of entanglement.

ToE synthesizes these developments into a coherent mechanism. Earlier generations simply did not have the conceptual machinery to make this leap.

6. The Inversion That No One Thought to Attempt

Perhaps the deepest reason is this: no one thought to invert the relationship between information geometry and physics. For decades, information geometry was used to analyze physical systems. It was never considered that physical systems might be emergent from information geometry.

This inversion is the heart of ToE:

Information geometry is not a tool for physics.  

Physics is a manifestation of information geometry.

Once this inversion is made, the pieces fall into place with startling clarity. But until the inversion is made, the connection remains invisible.

7. The Conceptual Leap That Changes Everything

The Theory of Entropicity succeeds because it makes a leap that earlier frameworks were not prepared to make. It treats information as ontological, not epistemic. It unifies classical and quantum information geometry. It promotes the Fisher–Rao metric, the Fubini–Study metric, and the alpha connections to physical status. It interprets their curvature as the curvature of reality. And it derives spacetime as an emergent projection of this deeper geometry.

This is why no one discovered it earlier. The pieces were scattered across disciplines, hidden behind assumptions, and separated by conceptual walls. ToE is the first framework to gather them, reinterpret them, and reveal the architecture they form together.