How Obidi Transformed Information Geometry into Physical Spacetime in the Theory of Entropicity (ToE)
1. Introduction
John Onimisi Obidi’s Theory of Entropicity (ToE) presents a radical reinterpretation of information geometry, transforming it from a mathematical framework for statistical inference into the ontological geometry of physical reality. In ToE, the structures traditionally used to quantify statistical distinguishability—such as the Fisher–Rao metric, the Fubini–Study metric, and the Amari–Čencov α‑connections—are reinterpreted as the pre‑spacetime geometric structures from which physical spacetime, matter, and interactions emerge.
This transformation is achieved through a sequence of conceptual and mathematical identifications that elevate entropy and distinguishability to the status of fundamental physical entities [1].
2. The Ontological Shift: Entropy as Fundamental Reality
In conventional physics, entropy is a derived quantity—a measure of disorder, ignorance, or coarse‑graining. Obidi overturns this view by asserting that entropy is ontological, not epistemic. The entropic/statistical manifold is not a mathematical convenience but the underlying manifold of reality itself.
Obidi introduces a fundamental scalar field ( S(x) ), the entropic field, whose gradients, curvature, and dynamics generate all physical phenomena. In this view:
Entropy is not a descriptor of physical systems.
Entropy is the substance from which physical systems arise.
The entropic manifold is the true configuration space of the universe.
This ontological shift is the foundation of ToE [1].
3. Metric Identification: From Distinguishability to Physical Distance
Information geometry defines distance through statistical distinguishability. Two probability distributions are “far apart” if they are easy to tell apart statistically. The Fisher–Rao metric (classical) and the Fubini–Study metric (quantum) quantify this.
Obidi’s key insight is that distinguishability is not merely statistical—it is a geometric invariant. He identifies:
The Fisher–Rao metric as the pre‑spacetime metric of the real sector.
The Fubini–Study metric as the pre‑spacetime metric of the complex/matter sector.
The curvature of this information‑geometric manifold is declared to be identical to the curvature of physical spacetime in the thermodynamic limit [1].
Thus, four‑dimensional spacetime is a coarse‑grained projection of a deeper, higher‑dimensional entropic manifold.
This identification is the basis of the Curvature Transfer Theorem, which later recovers Einstein’s equations as emergent identities [3].
4. The Role of the α‑Connection
Information geometry possesses a one‑parameter family of affine connections, the Amari–Čencov α‑connections. Each α corresponds to a different statistical interpretation.
Obidi identifies the α = 0 connection as the physically relevant one because:
It is torsion‑free.
It is metric‑compatible.
These are precisely the defining properties of the Levi‑Civita connection in General Relativity.
Thus:
α = 0 connection ≡ Levi‑Civita connection of emergent spacetime.
This identification is central to the emergence of Einsteinian geometry from the entropic manifold [1].
5. Dynamic Generation: The Obidi Action and the Master Entropic Equation
ToE is not merely kinematic; it is dynamical. Obidi introduces the Obidi Action, a universal variational principle defined on the entropic manifold.
Varying this action yields the Master Entropic Equation (MEE), which plays the role of the entropic ancestor of Einstein’s field equations [1].
In ToE:
Geometry is not assumed.
Geometry is generated by the dynamics of the entropic field.
Spacetime curvature is the macroscopic limit of entropic curvature.
Thus, Einstein’s equations are not fundamental—they are emergent identities.
6. The Vuli‑Ndlela Integral
The Vuli‑Ndlela Integral is an entropy‑constrained action functional that:
Maximizes allowable entropy production.
Suppresses entropy‑destroying trajectories.
Enforces the Second Law at the geometric level.
It does not penalize entropy increase. It penalizes entropy reversal, i.e., paths that would require negative entropic flux or violate the monotonicity of distinguishability.
Function of the Integral
The Vuli‑Ndlela Integral:
Weights each path by its entropic admissibility.
Selects the entropic geodesic (the extremal entropy‑producing path).
Enforces causal ordering through entropic cones.
Encodes both reversible and irreversible dynamics.
Ensures that physical motion follows the Second Law‑consistent extremal trajectory.
Thus, the integral is the mathematical heartbeat of ToE, governing how the entropic field evolves and how spacetime emerges from that evolution [2].
7. Summary of the Transformation Information Geometry Concept Physical Spacetime Equivalent Entropic/Statistical Manifold Fundamental Ontological Manifold Fisher–Rao / Fubini–Study Metric Pre‑spacetime Metric α = 0 Affine Connection Levi‑Civita Connection Entropy Gradients / Curvature Gravity and Spacetime Curvature Distinguishability Limits Speed of Light (c)
8. Conclusion
Obidi’s Theory of Entropicity provides a framework in which the curvature of physical spacetime is not a primitive assumption but an emergent thermodynamic‑limit expression of curvature defined on an underlying entropic manifold [3].
Through the Curvature Transfer Theorem (CTT), Obidi demonstrates that the spacetime Riemann tensor is the pushforward of the information‑geometric Riemann tensor. Einstein’s field equations are therefore recovered as emergent identities, not fundamental laws.
This transformation—turning information geometry into physical geometry—constitutes one of the most radical reinterpretations of the foundations of physics in the modern era.
References
[2] The Unified Entropy–Geometry Framework of the Theory of Entropicity (ToE)