The Conceptual Leap of the Theory of Entropicity (ToE) — From Information Geometry to the Architecture of Reality with One Stroke of Insight - Canonical
1. Standing at the Edge of a New Ontology
There are moments in the history of ideas when a conceptual framework stops behaving like a mere interpretation and begins to reveal itself as a new ontology. These moments are rare, and when they occur, they produce a distinctive sensation: a kind of intellectual vertigo, a giddiness that signals the mind is adjusting to a new scale of thought. The Theory of Entropicity (ToE) belongs to this class of transformative frameworks.
For decades, structures such as the Fisher–Rao metric, the Fubini–Study metric, and the Amari–Čencov alpha connections have been treated as tools — mathematical devices used to analyze probability distributions, quantum states, or learning algorithms. But when one recognizes that these structures may not be tools at all, but the very architecture of reality itself, the ground beneath physics begins to shift. The mind senses that it is standing at the threshold of a new landscape.
This is not the feeling of confusion. It is the feeling of conceptual expansion. It is the same cognitive recalibration that occurred when Newton unified celestial and terrestrial motion, when Einstein fused space and time into a single geometric entity, and when quantum theory revealed the dual nature of matter and light. The Theory of Entropicity (ToE) asks us to take a similar step: to see the geometry of information generated from the entropic field not as a mathematical convenience, but as the geometry of being.
2. Dissolving the Key Misunderstanding
The first obstacle to understanding Obidi’s Theory of Entropicity (ToE) is a deeply ingrained assumption: the belief that Fisher–Rao, Fubini–Study, and the alpha connections are tools we apply to information. This assumption is so pervasive that it feels almost tautological.
ToE overturns this assumption.
It asserts that these structures are not tools applied to information. They are the geometry that information itself naturally carries. And if information is the substrate of physical reality, then the geometry of information is the geometry of reality.
This is the pivot. Once this shift is made, the rest of the theory unfolds with surprising coherence.
3. Why This Move Is Not Arbitrary
It is important to emphasize that ToE is not an act of mathematical whimsy. It does not take random statistical constructs and elevate them to physical status. Instead, it recognizes that if the universe is fundamentally informational, then the geometry of that informational substrate is not optional. It is uniquely determined.
Several facts converge here:
• The Fisher–Rao metric is the only metric invariant under all information‑preserving transformations.
• The Amari–Čencov alpha connections are the only torsion‑free affine connections compatible with that metric under Markov morphisms.
• The Fubini–Study metric is the unique quantum analogue of Fisher–Rao, arising from the geometry of pure quantum states.
These structures are not arbitrary. They are forced by the internal logic of information itself as arising from entropy (the Entropic Field). If information is fundamental, these geometries are the only ones that can exist.
4. How Informational Geometry Produces Physical Spacetime
Once one accepts that reality is an informational manifold emergent from the entropic field, then the mechanism by which spacetime emerges becomes clear. The steps are conceptually simple, though profound in implication.
4.1 Reality as an Informational Manifold
ToE begins by positing an underlying manifold, denoted here as the entropic manifold. Its points correspond to informational states, not to spatial locations or physical events.
4.2 The Metric Structure
If information is the substrate, the manifold must carry a metric. The only consistent candidates are the Fisher–Rao metric in classical regimes and the Fubini–Study metric in quantum regimes. These are not imposed; they arise naturally from the structure of information.
4.3 The Affine Structure
The manifold must also carry an affine connection. The only connections compatible with informational invariance are the Amari–Čencov alpha connections. These define how informational states relate, evolve, and curve.
4.4 Informational Curvature
The curvature of these connections is not the curvature of a statistical model. It is the curvature of the informational substrate itself. This curvature is ontological.
4.5 Emergence of Spacetime
When the entropic manifold is coarse‑grained, the effective geometry that emerges is a four‑dimensional spacetime equipped with a metric that satisfies Einstein‑like field equations. In this view, gravitational curvature is the macroscopic shadow of informational (entropic) curvature.
4.6 Einstein’s Relativity as a Limit of Informational (Entropic) Geometry
Einstein’s metric is not fundamental. It is the emergent projection of the Fisher–Rao and Fubini–Study metrics. Einstein’s connection is not fundamental. It is the coarse‑grained limit of the alpha connections. Einstein’s curvature is not fundamental. It is the large‑scale limit of informational curvature.
Thus, general relativity is contained within the geometry of (entropic) information. It is not replaced; it is explained and given further enhancement.
5. Why This Has Never Been Done Before
The conceptual leap of ToE becomes clearer when one surveys the existing literature. In every field where these geometric structures appear, they are used in a representational or algorithmic capacity.
• In statistics, the Fisher–Rao metric measures distances between models.
• In quantum mechanics, the Fubini–Study metric measures distances between states.
• In machine learning, the alpha connections guide optimization and learning dynamics.
No one has ever proposed that these are not tools but the actual geometric structures of the universe. No one has unified Fisher–Rao and Fubini–Study as one metric. No one has promoted the alpha connections to physical affine connections. No one has treated informational curvature as ontological. No one has derived spacetime as a projection of an informational manifold.
This is the conceptual leap of the Theory of Entropicity (ToE).
6. A Historical Analogy That Clarifies the Shift
To understand the magnitude of this shift, it helps to recall the transformation Einstein introduced. Before Einstein, time was a coordinate, space was a stage, and gravity was a force. Einstein revealed that spacetime is a geometric object and gravity is curvature.
ToE performs a similar inversion:
Before ToE, information geometry was a tool, entropy was a measure, and probability distributions were models. ToE asserts that:
- information is the substrate (emergent from entropy),
- entropy is a field (generator of the information geometry),
- Fisher–Rao and Fubini–Study are the metric of reality,
- the alpha connections are the affine structure of reality, and
- spacetime is the emergent projection of informational curvature arising from entropy (the entropic field).
This is why the mind feels destabilized upon a first confrontation with the Theory of Entropicity (ToE). It is witnessing the collapse of an old ontology and the emergence of a new one.
7. Closure
Thus, the Theory of Entropicity is indeed saying that Einstein’s relativity is contained within the Fisher–Rao metric, the Fubini–Study metric, and the Amari–Čencov alpha connections. Not metaphorically. Not analogically. But literally, through emergence and coarse‑graining.
In other words, Einstein’s geometry is the macroscopic limit of informational geometry generated by the entropic field. This is the birth of the Theory of Entropicity (ToE).
References
- Grokipedia
— Theory of Entropicity (ToE)
https://grokipedia.com/page/Theory_of_Entropicity - Grokipedia
— John Onimisi Obidi
https://grokipedia.com/page/John_Onimisi_Obidi - Google
Blogger — Live Website on the Theory of Entropicity (ToE)
https://theoryofentropicity.blogspot.com - GitHub
Wiki on the Theory of Entropicity (ToE): https://github.com/Entropicity/Theory-of-Entropicity-ToE/wiki
- 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 and
the Laws of Nature. Encyclopedia. Available online: https://encyclopedia.pub/entry/59188
(accessed on 07 February 2026).
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