On the Cross-Domain Mathematical and Conceptual Complexity of the Theory of Entropicity (ToE): Foundations of a New Physics for the 21st Century
The Theory of Entropicity (ToE), first formulated and further developed by John Onimisi Obidi in early 2025, begins from a simple but radical premise:
That the most fundamental ingredient of physical reality is entropy, understood not as disorder or randomness, but as the deep measure of distinguishability between physical states.
In this view, the universe is not built from particles, fields, or spacetime itself, but from the entropic information that allows one state of the world to be told apart from another. Everything else—geometry, matter, energy, motion—emerges from the structure and flow of this entropic information [and the associated gradients].To make such a claim scientifically meaningful, ToE must translate the abstract idea of “entropic information” into a precise mathematical object capable of generating the familiar structures of physics. This is where the theory becomes subtle and conceptually rich.
The mathematics [and concepts] of the Theory of Entropicity (ToE) is not difficult because it is filled with symbols; it is difficult because it asks us to rethink what the symbols mean. It asks us to see information not as something stored in computers or communicated in messages, but as a physical field that lives on a manifold, interacts with itself, and shapes the geometry of the universe.The first step in this translation is the recognition that information, itself constructed from entropy, has a geometry. This is the insight of information geometry, a field pioneered by Fisher, Rao, Čencov, and Amari. In information geometry, the “distance” between two probability distributions is not measured in meters or seconds, but in how distinguishable they are.
Two distributions that are easy to tell apart are “far apart”; two that are nearly identical are “close.” This distance is encoded in a geometric object called the information metric, which measures how sensitive a distribution is to changes in its parameters. In the Theory of Entropicity (ToE), Obidi employs this metric to become the seed of spacetime itself. But information geometry is originally Riemannian—it has no notion of time, no causal structure, no light cones, no distinction between past and future. ToE introduces a new ingredient: the entropy field, a scalar field defined at every point of spacetime. The gradient of this field—the direction in which entropy increases most rapidly—provides a natural arrow of time.
By using this gradient to deform the information metric, Obidi is able to construct ToE to perform what is known as the Lorentzian lift (via what is known as the Obidi Transformation): a transformation that converts the timeless geometry of information into the time‑oriented geometry of spacetime. This is the moment where information becomes gravity, because the curvature of this Lorentzian geometry is what we recognize as gravitational curvature.Yet geometry is only half of Einstein’s theory of General Relativity (GR). The other half is the stress–energy tensor (SET), the object that encodes mass, pressure, radiation, momentum, and stress. ToE must show not only how information curves spacetime, but also how information becomes the source of that curvature.
This is where Obidi's theory reaches its deepest conceptual innovation in modern theoretical physics. In physics, the stress–energy tensor always arises from the flow of momentum. Whether we are describing a gas, a beam of light, a fluid, or a scalar field, the structure is the same: the energy density, pressure, and stresses are all determined by how momentum is distributed and how it moves. The Theory of Entropicity (ToE) adopts this universal structure, but with a twist: instead of describing particles or radiation, it describes the flow of information.To do this, ToE represents information as a distribution on the cotangent bundle of spacetime. The cotangent bundle is a mathematical space that pairs every point in spacetime with every possible momentum at that point. In ordinary physics, this is the natural home of particles and radiation. In the Theory of Entropicity (ToE), Obidi reconstructs it to become the natural home of information.
The information distribution tells us how much information is present at each point, and how it is moving—its direction, intensity, and flow. Once information is represented this way, the rest follows from the universal mathematics of kinetic theory. The second moment of the distribution—the average of the product of momentum with itself—produces a symmetric tensor with exactly the structure of the stress–energy tensor. This is not an analogy or a metaphor; it is a mathematical identity. The components of this tensor correspond to energy density, pressure, momentum flux, and shear stress because that is what the second moment of any momentum distribution must represent. In this way, information becomes mass, pressure, radiation, and stress not by assumption, but by the intrinsic geometry of momentum space.
The elegance of Obidi's Theory of Entropicity (ToE) lies in how these jigsaw pieces all fit together. The entropy field shapes the geometry of the information manifold, which becomes spacetime through the Lorentzian lift. The flow of information through this geometry produces the stress–energy tensor through the second moment of its distribution. The curvature of spacetime responds to this tensor through the Einstein equations. And the entropy field itself evolves according to the dynamics encoded in the Obidi Action of the Theory of Entropicity (ToE), which unifies the geometric, kinetic, and constraint sectors into a single entropic framework.
What makes the mathematics of ToE complex is not the presence of equations, but the interdependence of its structures:
Information geometry shapes spacetime geometry; spacetime geometry shapes information flow; information flow shapes the stress–energy tensor; the stress–energy tensor shapes curvature; curvature shapes the entropy field; and the entropy field shapes information geometry.
Thus, Obidi shows that Lorentzian spacetime geometry emerges from information geometry via a controlled entropy‑gradient disformal transformation, and that the curvature of this emergent metric reproduces the Einstein gravity of General Relativity (GR).
The theory is thus a closed loop, a self‑consistent system in which entropic information and geometry continually generate and constrain one another. The Theory of Entropicity (ToE) is therefore not a theory about information; it is a theory of information emergent from entropy itself. It does not treat information as a property of matter, but as the substrate from which matter and geometry emerge. It does not treat entropy as a measure of ignorance, but as the fundamental field that gives rise to that information [and its distribution] componented [projected] into time, causality, and gravitational dynamics. It does not treat spacetime as a stage on which physics happens, but as a projection of a deeper entropic informational manifold.
This is why the mathematics of ToE feels both familiar and foreign. It uses the tools of differential geometry, kinetic theory, and statistical mechanics, but it uses them in a radically new way, with new meanings and new connections. Obidi asks us to see information not as something abstract, but as something physical—something that moves, flows, curves, and interacts. Something that can be measured, integrated, and transformed. Something that can become mass, pressure, radiation, and gravity.
In this sense, Obidi's Theory of Entropicity (ToE) is not merely a new theory; it is a new way of thinking about what physics is made of. It replaces the ontology of particles and fields with an ontology of entropy and information. It replaces the geometry of spacetime with the geometry of distinguishability. And it replaces the traditional division between matter and geometry with a unified entropic structure from which both emerge. This is the conceptual heart of the Theory of Entropicity (ToE): the universe is the geometry of entropic information, and gravity is the curvature of that geometry.
References
1) https://doi.org/10.13140/RG.2.2.14211.26405
2) https://doi.org/10.17605/OSF.IO/PT9U8
4) Canonical Archive: https://entropicity.github.io/Theory-of-Entropicity-ToE/