How the Theory of Entropicity (ToE) Reimagines Fisher–Rao, Fubini–Study, and Amari–Čencov Geometry as the Foundations of Physical Reality

How the Theory of Entropicity (ToE) Reimagines Fisher–Rao, Fubini–Study, and Amari–Čencov Geometry as the Foundations of Physical Reality: Bedrock of the Obidi Action and Obidi Field Equations (OFE) of ToE 

For decades, the Fisher–Rao metric, the Fubini–Study metric, and the Amari–Čencov α‑connections have been central tools in information geometry, statistics, quantum theory, and machine learning in the computational science of Artificial Intelligence and Data Science. They have shaped how we understand probability distributions, quantum states, and learning algorithms. Yet in all these fields, these geometric structures have been used in a very specific way: they describe the geometry of models—the geometry of our representations of systems, not the geometry of the physical world itself.

The Theory of Entropicity (ToE) proposes a radical shift. It argues that these same geometric structures are not merely tools for analyzing data or optimizing algorithms. Instead, they form the underlying geometry of reality itself. In ToE, information is not a descriptor of physical systems; it is the substrate from which physical systems emerge. The geometry of information becomes the geometry of the universe.

This paper explains how ToE employs these mathematical structures in ways fundamentally different from their traditional uses, and why this shift represents a new direction in foundational physics.

From Statistical Manifolds to the Entropic Manifold

In classical information geometry, one studies statistical manifolds: spaces whose points represent probability distributions or density matrices. The geometry of these spaces tells us how distinguishable two distributions are, how learning algorithms behave, or how quantum states evolve. These manifolds are epistemic—they describe our knowledge about a system.

The Theory of Entropicity introduces a different kind of manifold: the entropic manifold. This is not a space of models. It is the informational substrate of reality itself. Points on this manifold correspond to primitive informational configurations, not to probability distributions chosen by an observer. The geometry of this manifold is not a geometry of inference; it is the geometry of existence.

This momentous leap—from epistemic geometry to ontological geometry—is the foundation on which ToE is built.

Fisher–Rao as the Physical Metric of Reality

Traditionally, the Fisher–Rao metric measures how easily two probability distributions can be distinguished. It is used to define thermodynamic length, natural gradient descent, and the geometry of statistical models. But it is always a metric on a space of models.

In ToE, the Fisher–Rao metric is reinterpreted as the physical metric of the entropic manifold. Instead of measuring distances between probability distributions, it measures distances between informational states of reality itself. The curvature of this metric is not a property of a model; it is a property of the universe.

Under coarse‑graining, this informational curvature gives rise to the familiar curvature of spacetime described by general relativity. In this view, Einstein’s geometry is not fundamental. It is an emergent, macroscopic shadow of a deeper informational geometry.

Fubini–Study as the Quantum Face of the Same Geometry

In quantum mechanics, the Fubini–Study metric measures the distinguishability of pure quantum states. It is defined on projective Hilbert space and plays a central role in geometric quantum mechanics. But again, it is a metric on a state space, not on spacetime.

The Theory of Entropicity unifies the Fisher–Rao and Fubini–Study metrics as two regimes of a single entropic geometry. The classical informational geometry (Fisher–Rao) and the quantum informational geometry (Fubini–Study) are not separate structures. They are different manifestations of the same underlying metric on the entropic manifold.

In this unified picture:

- The Fisher–Rao metric describes the classical limit of informational geometry.

- The Fubini–Study metric describes its quantum refinement.

- Both arise from the same entropic substrate.

This unification is not present in traditional information geometry, where the two metrics live on different spaces and serve different purposes. In the Theory of Entropicity (ToE), they are two faces of one geometry.

Amari–Čencov α‑Connections as Physical Affine Structure

The Amari–Čencov α‑connections are a family of affine connections used to study statistical models, generalized entropies, and learning algorithms. They define how one moves across a space of probability distributions or density matrices. Their curvature describes the geometry of inference, not the geometry of the universe.

In ToE, these α‑connections are elevated to the status of physical affine connections on the entropic manifold. Their curvature is interpreted as physically real informational curvature. Different values of α correspond to different physical regimes or phases of the entropic field.

This reinterpretation transforms the α‑connections from tools of statistical analysis into the actual connection coefficients of the informational universe. Their curvature enters directly into the fundamental field equations of ToE, just as the Levi‑Civita connection enters Einstein’s field equations of General Relativity (GR).

Entropy as a Field That Shapes the Geometry of Reality

In traditional physics, entropy is a measure of disorder or uncertainty. In information geometry, it is a functional on probability distributions. In ToE, entropy becomes something far more fundamental: an autonomous field defined on the entropic manifold.

This entropic field interacts with the geometry of the manifold. Its gradients and fluxes act as sources of informational curvature. The resulting curvature, when viewed through the lens of coarse‑graining, appears as gravitational curvature in the emergent spacetime.

In this way, ToE proposes a new kind of field equation—an informational analogue of Einstein’s equations—where entropy plays the role of a source term.

A New Paradigm: From Geometry of Models to Geometry of Reality

The key distinction between ToE and traditional uses of information geometry can be summarized simply:

- Traditional information geometry studies the geometry of models, distributions, and states of knowledge.

- The Theory of Entropicity (ToE) studies the geometry of reality itself, treating information as the fundamental substrate.

In the Theory of Entropicity (ToE):

- The Fisher–Rao metric becomes the physical metric of the informational universe.

- The Fubini–Study metric becomes the quantum refinement of that same geometry.

- The Amari–Čencov α‑connections become the physical affine structure of the entropic manifold.

- Entropy becomes a field whose dynamics shape the curvature of reality.

This is not a reinterpretation of existing mathematics. It is a re‑anchoring of those mathematical structures in the ontology of the physical world.

Why This Matters for a Foundational Principle of Nature 

If the geometry of information is the geometry of reality, then the divide between classical physics, quantum mechanics, and gravity (Einstein's Relativity) is not a divide at all. They are different regimes of a single informational field theory. In this way, the Theory of Entropicity (ToE) thus offers a unified geometric language that spans these domains, grounded in structures that have been studied for decades but never given ontological status.

This singular, elegant and innovative shift by the Theory of Entropicity (ToE)—from using information geometry as a tool to recognizing it as the foundation of physical law—opens the door to a new way of understanding the universe. It suggests that the deepest structures of physics are informational, not material, and that spacetime itself, as well as gravity, is an emergent phenomenon arising from the curvature of an underlying entropic manifold as formulated in Obidi's audacious Theory of Entropicity (ToE).

References

1. Grokipedia — Theory of Entropicity (ToE)
   https://grokipedia.com/page/Theory_of_Entropicity
2. Grokipedia — John Onimisi Obidi
   https://grokipedia.com/page/John_Onimisi_Obidi
3. Google Blogger — Live Website on the Theory of Entropicity (ToE)
   https://theoryofentropicity.blogspot.com
4. GitHub Wiki on the Theory of Entropicity (ToE): https://github.com/Entropicity/Theory-of-Entropicity-ToE/wiki
5. Canonical Archive of the Theory of Entropicity (ToE)
   https://entropicity.github.io/Theory-of-Entropicity-ToE/
6. John Onimisi Obidi. Theory of Entropicity (ToE): Path To Unification of Physics. Encyclopedia. Available online: https://encyclopedia.pub/entry/59188 (accessed on 07 February 2026).