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The Information‑Geometry Bridge of the Theory of Entropicity (ToE)

From Fisher–Rao and Fubini–Study Metrics to α‑Connections, Entropic Curvature, and the Geometry of Physical Spacetime

This paper presents a unified exposition of the Information‑Geometry Bridge at the heart of the Theory of Entropicity (ToE)physical spacetime emerges from entropy dynamics.

The Information‑Geometry Bridge asserts that the geometry of physical spacetime is not fundamental. Instead, it is an effective, coarse‑grained manifestation of a deeper information‑geometric manifold governed by the entropic field S(x). This field is continuous, dynamical, and defined on a differentiable manifold. Its gradients, curvature, and thresholded distinguishability structure generate gravity, motion, time dilation, and the causal architecture of the universe.

The Information‑Geometry Bridge is an emerging theoretical framework that attempts to unify physical spacetime geometry with statistical information flow. It leverages information geometry—the differential geometry of probability distributions—to model physical space, motion, and gravity as emergent phenomena arising from entropy gradients and the reconfiguration or redistribution of information (entropy).

Within this landscape, the Theory of Entropicity (ToE) provides the most direct and explicit realization of this bridge. It elevates entropy from a statistical byproduct to a fundamental, dynamic field and uses information‑geometric tools to reconstruct spacetime, gravitation, and relativistic effects from entropic principles.

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1. The Theory of Entropicity (ToE)

The Theory of Entropicity (ToE), developed by John Onimisi Obidi, explicitly utilizes information‑geometric tools to redefine physical reality in terms of entropy and its geometry.

1.1 Core Principle

The core principle of ToE is:

Entropy is a fundamental, dynamic field S(x) whose gradients generate spacetime geometry, gravity, time dilation, and motion.

Here, S(x) is defined on a differentiable manifold and is treated as a real, local, dynamical field. Physical phenomena—such as gravitational curvature, inertial motion, and relativistic effects—are interpreted as manifestations of the geometry and dynamics of this entropic field.

1.2 Mathematical Tools in ToE

ToE combines several key information‑geometric structures:

• Fisher–Rao metric: quantifies classical statistical distinguishability between probability distributions.
• Fubini–Study metric: quantifies quantum distinguishability between pure quantum states.
• Hybrid Metric‑Affine Space (HMAS): a unified structure in which Fisher–Rao and Fubini–Study metrics coexist and jointly determine the geometry of the entropic manifold.
• Amari–Čencov α‑connections: affine connections used to describe dualistic structures (mixture and exponential families) and the curvature of information flow.

These ingredients are combined into a single geometric framework in which information‑geometric curvature is identified with physical spacetime curvature.

1.3 The Obidi Action and the Master Entropic Equation

ToE defines an action principle known as the Obidi Action. This action generates the Master Entropic Equation (MEE), also referred to as the Obidi Field Equations (OFE).

Conceptually, the OFE replace Einstein’s field equations (EFE) with an equation of the form:

information‑geometric curvature = physical spacetime curvature

In this view, the curvature that appears in gravitational phenomena is not fundamental but is induced by the underlying information‑geometric curvature of the entropic field S(x).

1.4 Key Physical Results in ToE

Within this framework, ToE derives standard relativistic effects as consequences of entropic dynamics:

• Time dilation
• Length contraction
• Relativistic mass increase

These effects arise from entropic resistance and the existence of a maximum rate of information transfer, denoted IT, which is identified with the constant c (numerically equivalent to the speed of light in Einstein’s Theory of Relativity).

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2. Foundational Mathematical Frameworks

Several foundational works in mathematical physics connect information metrics directly to geometric structures. These provide the mathematical backbone for the Information‑Geometry Bridge and for ToE.

2.1 Amari–Čencov Theorem and α‑Connections

The Amari–Čencov theorem establishes that the Fisher–Rao metric and the associated α‑connections are uniquely invariant under sufficient statistics. This result is central to information geometry: it identifies the Fisher–Rao metric as the canonical Riemannian metric on the space of probability distributions and the α‑connections as the canonical affine structures compatible with it.

Modern research has shown that the α‑connections are, in fact, the Levi‑Civita connections of specifically defined Riemannian metrics on the space of density functions. These metrics are often referred to as α‑Fisher–Rao metrics.

2.2 α‑Fisher–Rao Metrics

The classical Fisher–Rao metric has been generalized to a family of α‑Fisher–Rao metrics, each associated with a particular α‑connection. These generalized metrics:

• encode different aspects of information geometry,
• are directly linked to α‑connections for various values of α,
• are used to study geodesic equations representing optimal paths of information flow.

In the context of ToE, these α‑Fisher–Rao metrics provide a natural way to describe how information flows and curves in the entropic manifold, and how this curvature is mapped to physical spacetime.

2.3 G‑Dual Teleparallel Pairs

Ciaglia and collaborators have developed a framework using Jordan algebras to combine the Fisher–Rao and Fubini–Study metrics in both classical and quantum contexts. This leads to the notion of G‑dual teleparallel pairs, which:

• unify classical statistical manifolds and quantum state manifolds,
• provide a joint description of classical and quantum information geometry,
• offer a natural setting for ToE’s Hybrid Metric‑Affine Space (HMAS).

These constructions show how classical and quantum information metrics can be treated within a single geometric framework, which is precisely what ToE exploits to connect information flow with spacetime geometry.

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3. Key Concepts in the Information‑Geometry Bridge of ToE

3.1 Information as Geometry

A central idea in the Information‑Geometry Bridge is that information is geometry. The structure of probability distributions, their distinguishability, and their flows can be represented as geometric objects—metrics, connections, and curvature—on statistical manifolds.

Within ToE, this leads to the notion of the "temperature of geometry": rapid informational change corresponds to a “hotter,” more dynamically curved spacetime, while slowly varying information corresponds to a “colder,” more static geometry.

3.2 The No‑Rush Theorem

The No‑Rush Theorem is a principle that enforces a lower bound on causal intervals. It states that no physical transition in the entropic field can be completed in zero time. This theorem:

• links information‑theoretic limits to physical spacetime structure,
• ensures that causal processes have finite duration,
• acts as a bridge between entropic dynamics and relativistic causality.

3.3 Rényi–Tsallis α–q Parameters

ToE employs Rényi and Tsallis generalized entropies to capture non‑extensive and non‑linear aspects of information. The parameters α (from α‑connections) and q (from Tsallis entropy) are related in such a way that:

q is mapped to α, connecting non‑extensive entropy deformation to geometric deformation.

This relation allows ToE to interpret changes in q as changes in the underlying affine and metric structure of the entropic manifold, thereby linking generalized entropy to spacetime deformation.

The Information‑Geometry Bridge, as developed in ToE and related works, is an actively evolving field. It appears across multiple academic repositories (TechRxiv, ResearchGate, arXiv, etc.) and is part of a broader effort to reconcile general relativity with quantum mechanics via informational and entropic frameworks.

4. Appendix: Extra Matter I — Extended Foundations of the Information‑Geometry Bridge

A wide range of contemporary research programs utilize the Amari–Čencov framework, Fisher–Rao and Fubini–Study metrics, and generalized entropy measures to build a bridge between information geometry and the geometry of physical spacetime. These works collectively form the intellectual backdrop against which the Theory of Entropicity (ToE) was developed.

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5. The Theory of Entropicity (ToE) in the Broader Information‑Geometric Landscape

Among all modern approaches, the Theory of Entropicity is the most direct and comprehensive framework linking:

• Amari–Čencov α‑connections
• Fisher–Rao and Fubini–Study metrics
• generalized entropies (Rényi, Tsallis)
• affine and Riemannian geometry
• physical spacetime curvature

ToE treats entropy as the fundamental field whose gradients generate gravitation, time, and motion. This is not metaphorical: the entropic field S(x) is the ontological substrate from which spacetime emerges.

5.1 The Bridge Mechanism

ToE uses the Amari–Čencov α‑connections to link:

• irreversibility of information flow
• affine asymmetry in the entropic manifold
• the deformation of spacetime geometry

The Fisher–Rao metric provides the classical distinguishability structure, while the Fubini–Study metric provides the quantum distinguishability structure. Together, they form a unified informational metric that governs the curvature of spacetime.

5.2 Entropic Lorentz Factor

ToE re‑derives the Lorentz factor γ of special relativity as a special case of a more general entropic Lorentz factor, which arises from the geometry of the dual α‑connections. This entropic Lorentz factor encodes:

• time dilation
• length contraction
• relativistic mass increase

These effects are interpreted as consequences of entropy conservation and the maximum rate of information transfer.

5.3 Constitutive Relation Between q and α

A key mechanism in ToE is the constitutive relation linking:

non‑extensive entropy deformation (q) ↔ affine asymmetry (α)

This relation forms the geometric bridge between generalized entropy and spacetime curvature.

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6. Ariel Caticha’s “Information Geometry of Space‑Time”

Ariel Caticha’s work provides one of the earliest and clearest demonstrations that physical space can be modeled as a statistical manifold.

6.1 Blurred Space

Caticha models physical space as a collection of points with finite resolution — a “blurred space.” The metric of this space is automatically the Fisher information metric.

6.2 Geometrodynamics

By requiring that this information‑geometric space sweeps out a 4‑dimensional manifold during evolution, Caticha shows that the resulting geometrodynamics reproduce:

Einstein’s vacuum field equations

This is a major conceptual breakthrough: it shows that general relativity can emerge from information geometry alone.

6.3 Limitations

Caticha notes that informational metrics are positive‑definite (Riemannian), while physical spacetime is Lorentzian. Additional structure is required to recover the light‑cone geometry.

ToE provides this missing structure through:

• entropic curvature
• thresholded distinguishability
• dual α‑connections

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7. Geometric Quantum Mechanics and Information Geometry (Ciaglia et al.)

Ciaglia and collaborators study the geometry of quantum state space using the Fubini–Study metric and the Quantum Fisher Information metric.

7.1 G‑Dual Teleparallel Pair

They introduce the G‑dual teleparallel pair, a structure that unifies:

• classical information geometry (Fisher–Rao)
• quantum geometry (Fubini–Study)

This framework shows how classical and quantum information metrics can be treated as dual aspects of a single geometric structure — a key insight used in ToE’s Hybrid Metric‑Affine Space.

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8. Riemannian Viewpoint on Amari–Čencov α‑Connections (2025)

Recent work provides a new geometric interpretation of the Amari–Čencov α‑connections by showing that they are the Levi‑Civita connections of a family of Riemannian metrics known as α‑Fisher–Rao metrics.

This result bridges the gap between:

• the affine structures of early information geometry
• metric‑based gravity in general relativity

It provides a rigorous mathematical foundation for ToE’s claim that spacetime curvature is induced by information‑geometric curvature.

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9. Core Concepts in the Information‑Geometry Bridge

9.1 Information Flow (Amari–Čencov)

The α‑connections describe how information flows along curved statistical manifolds. They encode:

• irreversibility
• dualistic structure (mixture vs exponential families)
• optimal information‑flow paths

9.2 Fisher–Rao Metric

The Fisher–Rao metric quantifies distinguishability between classical probability distributions. In the Information‑Geometry Bridge, it maps to the spatial metric of physical space.

9.3 Fubini–Study Metric

The Fubini–Study metric quantifies distinguishability between quantum states. It accounts for quantum complexity and the geometry of quantum evolution.

9.4 Physical Spacetime Geometry

In the Information‑Geometry Bridge, physical spacetime geometry is interpreted as the Levi‑Civita connection of an informational metric. This aligns with:

• entropic gravity (Verlinde)
• thermodynamic gravity (Jacobson)
• ToE’s entropic curvature framework

The conclusion is clear:

Einstein’s metric is not fundamental. It is an emergent, coarse‑grained representation of an underlying information‑geometric manifold.

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10. Appendix: Extra Matter II — Additional Frameworks in the Information‑Geometry Bridge

Beyond the Theory of Entropicity (ToE), several major research programs use information geometry— particularly the Amari–Čencov α‑connection framework and the Fisher–Rao / Fubini–Study metrics—to construct bridges between information flow and the geometry of physical spacetime. These works collectively reinforce the central thesis of the Information‑Geometry Bridge: that spacetime geometry is an emergent, informational construct.

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11. The Theory of Entropicity (ToE) — Extended Technical Integration

ToE integrates the Fisher–Rao and Fubini–Study metrics through the α‑connection formalism to derive physical phenomena directly from entropic geometry.

11.1 Mechanism of Integration

ToE treats entropy as a fundamental field whose gradients generate:

• motion
• gravitation
• time dilation
• mass increase
• length contraction

These effects arise from the geometry of the dual α‑connections and the entropic curvature of the manifold.

11.2 Entropic Lorentz Factor

The traditional Lorentz factor γ is reformulated as an entropic Lorentz factor, derived from the informational geometry of the α‑connections. This provides a deeper explanation for relativistic effects as consequences of entropy conservation and the maximum rate of information transfer.

11.3 Constitutive Relation Between q and α

ToE introduces a constitutive relation linking:

non‑extensive entropy deformation (q) ↔ affine asymmetry (α)

This relation forms the geometric bridge between generalized entropy (Tsallis/Rényi) and the information‑geometric curvature that manifests as physical spacetime curvature.

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12. Ariel Caticha’s Information Geometry of Space‑Time

Caticha’s framework is one of the earliest attempts to derive spacetime geometry from information geometry using the method of maximum entropy.

12.1 Blurred Statistical Manifold

Physical space is modeled as a “blurred” statistical manifold, where each point has finite resolution. The metric of this blurred space is automatically the Fisher information metric.

12.2 Emergence of Einstein’s Equations

By requiring that this statistical manifold evolves in time to sweep out a 4‑dimensional structure, Caticha shows that the resulting geometrodynamics reproduce:

Einstein’s vacuum field equations

12.3 Limitations

Because informational metrics are positive‑definite (Riemannian), additional structure is needed to recover Lorentzian spacetime. ToE provides this structure through:

• entropic curvature
• thresholded distinguishability
• dual α‑connections

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13. Geometry–Information Duality (Neukart et al.)

Neukart and collaborators propose that time arises as a local informational field rather than a universal coordinate.

13.1 Information Flow as Temporal Direction

Temporal direction is identified with gradients in stored information. This links:

• entropy gradients
• curvature
• the arrow of time

13.2 Emergent Lorentzian Geometry

In regions where informational gradients vary slowly, the emergent geometry reduces to the familiar Lorentzian structure of general relativity.

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14. Geometric Information Flows (Sergiu Vacaru)

Vacaru develops nonholonomic W‑entropy functionals to describe the evolution of relativistic mechanical systems.

14.1 W‑Entropy and Nonholonomic Geometry

These functionals describe geometric flows that evolve according to entropy‑based principles. Vacaru’s work studies how these flows project onto Lorentzian spacetime manifolds.

14.2 Entropic Force Interpretation

Vacaru’s framework suggests that gravitational field equations emerge from geometric flows characterized by entropy, aligning with:

• entropic gravity (Verlinde)
• thermodynamic gravity (Jacobson)
• ToE’s entropic curvature

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15. Key Mathematical Components in the Information‑Geometry Bridge

15.1 Fisher–Rao Metric

The Fisher–Rao metric acts as the primary Riemannian metric on the space of probability distributions. In many frameworks, it maps directly to the spatial metric of physical space.

15.2 Fubini–Study Metric

The Fubini–Study metric bridges quantum state space to classical information geometry. It is essential for describing quantum complexity and quantum evolution.

15.3 Amari–Čencov α‑Connections

The α‑connections provide the dualistic structure necessary to describe:

• irreversible field behavior
• energy‑minimizing paths
• curved informational manifolds

These connections are central to ToE’s interpretation of spacetime as an entropic manifold.

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16. Final Synthesis: The Information‑Geometry Bridge as a Unified Framework

Across all these frameworks—ToE, Caticha, Ciaglia, Neukart, Vacaru—the same conclusion emerges:

Spacetime geometry is not fundamental. It is an emergent, coarse‑grained representation of an underlying information‑geometric manifold.

The Theory of Entropicity provides the most complete realization of this idea by:

• treating entropy as a fundamental field
• using Fisher–Rao and Fubini–Study metrics to define informational curvature
• using α‑connections to encode irreversible information flow
• deriving spacetime curvature from entropic curvature
• introducing thresholded distinguishability (ln(2)) to explain quantization
• replacing Einstein’s field equations with the Obidi Field Equations

The Information‑Geometry Bridge is therefore not merely a mathematical analogy. It is a structural unification of:

• information theory
• geometry
• entropy
• quantum mechanics
• general relativity

In this unified view, the universe is an entropic manifold whose geometry encodes all physical phenomena. Spacetime, gravity, motion, and quantum behavior emerge from the geometry of entropy itself.

Extra Matter

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1. Entropy as a Fundamental Field

The Theory of Entropicity begins with a single axiom:

S(x) is a real, continuous, dynamical entropic field defined on a differentiable manifold M.

Its evolution is governed by the Obidi Action:

I[S] = ∫ L(S, ∇S, ∇²S, …) dμ

This action generates the Obidi Field Equations (OFE), also called the Master Entropic Equation (MEE). These equations replace Einstein’s field equations by asserting that:

entropic curvature = physical spacetime curvature

Thus, spacetime geometry is not fundamental; it is induced by the entropic field’s information‑geometric structure.

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2. Classical and Quantum Distinguishability Metrics

2.1 Fisher–Rao Metric (Classical)

The Fisher–Rao metric measures the distinguishability of classical probability distributions. It is the unique Riemannian metric invariant under sufficient statistics. In ToE, it forms the classical component of the hybrid information‑geometric structure.

2.2 Fubini–Study Metric (Quantum)

The Fubini–Study metric measures the distinguishability of quantum states on projective Hilbert space. It is the geometric backbone of quantum mechanics and quantum complexity.

2.3 Hybrid Metric‑Affine Space (HMAS)

ToE unifies these two metrics into a single Hybrid Metric‑Affine Space (HMAS), where classical and quantum distinguishability coexist and jointly determine the curvature of the entropic manifold.

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3. Amari–Čencov α‑Connections and the Geometry of Information Flow

Information geometry is built on the dualistic structure of α‑connections, which describe how information flows and curves on statistical manifolds. These connections capture the asymmetry between mixture and exponential families of distributions.

ToE uses α‑connections to describe:

• irreversible information flow
• entropic deformation of spacetime
• the emergence of affine asymmetry
• the geometric structure underlying entropic curvature

Recent mathematical results show that α‑connections are the Levi‑Civita connections of a family of Riemannian metrics called α‑Fisher–Rao metrics. This provides a direct bridge from information geometry to physical geometry.

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4. Generalized Entropies: Rényi, Tsallis, and Non‑Extensive Geometry

ToE incorporates generalized entropy measures — particularly Rényi and Tsallis entropies — to capture non‑extensive, nonlinear, and long‑range informational effects.

The key relation is the mapping between the Tsallis parameter q and the α‑connection parameter α:

q ↔ α

This mapping allows ToE to interpret non‑extensive entropy deformation as geometric deformation of the entropic manifold. As q deviates from 1, the geometry becomes more curved, encoding gravitational and relativistic effects.

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5. The No‑Rush Theorem and the Minimum Entropic Threshold

Two structural principles govern the emergence of physical events:

5.1 No‑Rush Theorem (NRT)

No entropic transition can be completed in zero time.

5.2 Obidi Curvature Invariant (OCI)

A physical distinction is realized only when entropic deformation exceeds ln(2).

Together, these yield the Integrated Entropic Interaction Measure (IEIM):

I_Ω[S] = ∫(over Ω) ρ_ent[S] dμ

A physical event occurs only if:

I_Ω[S] ≥ ln(2)

This thresholded structure explains why continuous entropic evolution produces discrete, quantized physical events.

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6. The Sector Index and Quantized Event Realization

Define the local entropic curvature κ[S]. Then the sector index is:

N(x) = floor( κ[S] / ln(2) )

This index changes only when κ[S] crosses integer multiples of ln(2). Thus:

• the entropic field evolves continuously
• the curvature evolves continuously
• but distinguishability sectors change discretely

This is the entropic origin of quantization.

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7. From Information Geometry to Physical Spacetime

The Information‑Geometry Bridge asserts that:

Physical spacetime is the macroscopic shadow of an underlying information‑geometric manifold.

Key consequences:

• gravity emerges from entropic curvature
• time dilation arises from entropic resistance
• motion is the geodesic flow of information
• the speed of light is the maximum rate of information transfer

This aligns with and extends the work of Caticha, Ciaglia, Vacaru, and others, but ToE provides the first fully unified, thresholded, dynamical formulation.

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8. The Aharonov–Bohm Effect in the Context of ToE

The AB effect demonstrates that global gauge potentials have physical consequences even in field‑free regions. ToE explains this by showing that:

• global gauge holonomy is encoded in entropic curvature
• the entropic field carries the topological information
• interference shifts occur when entropic thresholds are crossed

Thus, the AB effect is not an anomaly — it is a natural consequence of entropic geometry.

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9. Conclusion

The Information‑Geometry Bridge of the Theory of Entropicity unifies:

• classical information geometry (Fisher–Rao)
• quantum geometry (Fubini–Study)
• generalized entropies (Rényi, Tsallis)
• α‑connections and dualistic structure
• Riemannian and affine geometry
• entropic curvature and thresholded distinguishability

The result is a single, coherent framework in which:

Spacetime, gravity, motion, and quantization emerge from the geometry of entropy itself.

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