Entropy as the Fundamental Field: A Unified Geometric and Spectral Foundation for the Theory of Entropicity (ToE)

Abstract

The Theory of Entropicity (ToE) proposes a radical restructuring of modern physics by elevating entropy from a derived statistical descriptor to the fundamental field and ontological substrate of the universe. This paper integrates recent advances in information geometry—particularly the Riemannian interpretation of the Amari–Čencov α‑connections—into the mathematical foundation of ToE. By demonstrating that α‑connections arise as Levi‑Civita connections of α‑Fisher–Rao metrics, and by connecting these structures to hydrodynamic PDEs such as the Proudman–Johnson and Hunter–Saxton equations, we establish a rigorous geometric and variational framework for the Obidi Action, the Local Obidi Action (LOA), and the Spectral Obidi Action (SOA). The result is a unified entropic field theory in which geometry, quantum behavior, and physical dynamics emerge from the intrinsic structure and evolution of entropy itself.

1. Introduction: Entropy as the Ontic Substrate of Reality

The Theory of Entropicity (ToE) begins with a conceptual inversion: entropy is not a measure of disorder, ignorance, or coarse‑graining, but the primary field from which all physical structures arise. In this view, the universe is not governed by geometric or energetic laws that subsequently give rise to entropy; instead, the universe evolves according to the dynamics of an entropic field, and geometry, matter, and interaction emerge as secondary expressions of this evolution.

Let $\mathcal{E}(x)$ denote the entropic field. The fundamental postulate of ToE asserts that all physical laws arise from the extremization of an entropic action functional

$$S_{\text{Obidi}}[\mathcal{E}]={\int }_{\mathcal{M}}\mathcal{L}(\mathcal{E},\mathrm{\nabla }\mathcal{E})d\mu ,$$

where $\mathcal{L}$ is an entropic Lagrangian constructed from information‑geometric and spectral quantities. The resulting Euler–Lagrange equation defines the Master Entropic Equation (MEE), the universal field equation of ToE.

2. Information Geometry as the Mathematical Backbone of ToE

Information geometry provides the natural mathematical language for ToE because it treats probability densities as points on a differentiable manifold equipped with canonical geometric structures. Two of these structures are central:

The Fisher–Rao metric on the space of densities ${\mathrm{D}\mathrm{e}\mathrm{n}\mathrm{s}}^{+}(M)$ is defined by

$$G_{\mathrm{F}\mathrm{R}}(a,b)={\int }_M\frac{ab}{\rho }d\mu ,$$

where $\rho \in {\mathrm{D}\mathrm{e}\mathrm{n}\mathrm{s}}^{+}(M)$ and $a,b\in T_{\rho }{\mathrm{D}\mathrm{e}\mathrm{n}\mathrm{s}}^{+}(M)$.

The Amari–Čencov α‑connections are defined by

$${\mathrm{\nabla }}_a^{(\alpha )}b=Db[a]-\frac{1+\alpha }{2}\frac{ab}{\rho },$$

where $Db[a]$ is the directional derivative of $b$ along $a$.

In classical information geometry, the α‑connections interpolate between the mixture connection (α$=-1$), the exponential connection ($\alpha =1$), and the Levi‑Civita connection of the Fisher–Rao metric ($\alpha =0$).

ToE extends this structure by interpreting the α‑connections as entropic connections, encoding how the entropic field curves, flows, and generates physical behavior.

3. The Riemannian Interpretation of α‑Connections and Its Significance for ToE

A major mathematical breakthrough comes from the result that for every real $\alpha$, there exists a Riemannian metric $G_{\alpha }$ whose Levi‑Civita connection is exactly the Amari–Čencov α‑connection. This metric is the α‑Fisher–Rao metric, defined by

$$G_{\alpha }(a,b)={\int }_M{\rho }^{-\alpha -1}abd\mu \mathrm{.}$$

This result has profound implications for ToE:

1. It proves that α‑connections are not merely statistical constructs but arise from legitimate Riemannian geometry.

2. It validates the use of α‑connections in the Obidi Action as geometric objects with well‑defined curvature and geodesics.

3. It provides a variational interpretation of entropic evolution, since α‑geodesics minimize the Riemannian energy

$$E_{\alpha }(t)=G_{\alpha }(\dot{\rho }(t),\dot{\rho }(t))\mathrm{.}$$

Thus, the entropic field evolves along α‑geodesics, giving ToE a rigorous geometric foundation.

4. The Obidi Action, LOA, and SOA in Light of α‑Geometry

The Local Obidi Action (LOA) is the geometric formulation of ToE, constructed from local entropic densities. Using the α‑Fisher–Rao metric, the LOA takes the form

$${\mathrm{S\_}}_{\mathrm{L}\mathrm{O}\mathrm{A}}[\rho ]={\int }_{\mathcal{M}}{G}_{\alpha }(\mathrm{\nabla }\rho ,\mathrm{\nabla }\rho )d\mu ,$$

which yields the α‑geodesic equation as its Euler–Lagrange equation.

The Spectral Obidi Action (SOA) lifts the theory into the operator‑algebraic domain. Let ${\mathrm{\Delta }}_{\rho \mathrm{\mid }\sigma }$ denote the relative modular operator associated with two states $\rho$ and $\sigma$. The Araki relative entropy is

$$S(\rho \mathrm{\parallel }\sigma )=-⟨\log {\mathrm{\Delta }}_{\rho \mathrm{\mid }\sigma }{⟩}_{\rho }\mathrm{.}$$

ToE promotes this quantity from a diagnostic tool to a generative principle by defining

$${\mathrm{S\_}}_{\mathrm{S}\mathrm{O}\mathrm{A}}[\rho ]={\int }_{\mathcal{M}}S(\rho (x)\mathrm{\parallel }\sigma (x))d\mu ,$$

where $\sigma$ is a reference entropic configuration.

The α‑geometry results justify this construction by showing that entropic divergences naturally generate geometric connections and geodesics.

5. Geodesics, α‑Root Maps, and the Flattening of Entropic Geometry

A remarkable result from the attached paper is that the α‑Fisher–Rao metric becomes flat under the α‑root map

$${\mathrm{\Phi }}_{\alpha }(\rho )=\{\begin{array}{cc}\frac{2}{1-\alpha }{\rho }^{\frac{1-\alpha }{2}},& \alpha \mathrm{\ne }1,\\ \log \rho ,& \alpha =1.\end{array}$$

Under this transformation, α‑geodesics become straight lines in function space. This supports the SOA’s use of modular operators to “flatten” entropic geometry, showing that entropy possesses a natural linearizing transformation.

6. Hydrodynamic PDEs as Entropic Geodesics: Proudman–Johnson and Hunter–Saxton

The attached paper demonstrates that α‑geodesics on the space of densities pull back to hydrodynamic PDEs on diffeomorphism groups. For example, the generalized Proudman–Johnson equation

$$u_{txx}+(2-\alpha )u_x{u}_{xx}+u{u}_{xxx}=0$$

arises as the Euler–Arnold equation of the α‑connection.

This provides a concrete example of the ToE principle that physical dynamics emerge from entropic geometry. It shows that entropic geodesics can generate nonlinear PDEs governing fluid motion, supporting the idea that Einstein’s equations, Yang–Mills dynamics, and quantum evolution may similarly arise from the MEE.

7. Entropic Duality and the α ↔ −α Symmetry

The α‑connections satisfy the duality relation

$${\mathrm{\nabla }}^{(\alpha )}\text{ is dual to }{\mathrm{\nabla }}^{(-\alpha )}\text{ under }{G}_{\mathrm{F}\mathrm{R}}\mathrm{.}$$

This duality mirrors the LOA–SOA duality in ToE:

LOA corresponds to local geometric evolution. SOA corresponds to spectral evolution under modular operators.

The α‑geometry results provide a rigorous mathematical foundation for this duality.

8. Conclusion: A Unified Entropic Field Theory

The integration of α‑geometry into the Theory of Entropicity yields a coherent and mathematically rigorous framework in which entropy is the fundamental field of the universe. The α‑Fisher–Rao metrics, α‑connections, and their Riemannian interpretation provide the geometric structure needed for the Obidi Action. The spectral properties of Araki relative entropy provide the operator‑theoretic structure needed for the SOA. The emergence of hydrodynamic PDEs from entropic geodesics demonstrates the physical generativity of the framework.

ToE thus stands as a unified entropic field theory in which geometry, quantum mechanics, thermodynamics, and physical dynamics all arise from the intrinsic structure and evolution of entropy itself.

Appendix - Some Extra Matter

4.1 Introduction: The Entropic Reversal of Physical Ontology

The Theory of Entropicity (ToE) proposes a conceptual inversion that challenges the deepest assumptions of modern physics. In classical thermodynamics, statistical mechanics, and even quantum information theory, entropy is treated as a secondary descriptor—an emergent measure of disorder, uncertainty, or coarse‑graining. ToE rejects this epistemic interpretation and instead asserts that entropy is ontic: the fundamental field from which geometry, matter, quantum behavior, and dynamical law arise.

Let $\mathcal{E}(x)$ denote the entropic field defined over a differentiable manifold $\mathcal{M}$. The central postulate of ToE is that the universe evolves according to an entropic variational principle, encoded in the Obidi Action

$$S_{\mathrm{O}\mathrm{b}\mathrm{i}\mathrm{d}\mathrm{i}}[\mathcal{E}]={\int }_{\mathcal{M}}\mathcal{L}(\mathcal{E},\mathrm{\nabla }\mathcal{E})d\mu ,$$

where $\mathcal{L}$ is an entropic Lagrangian constructed from information‑geometric and spectral quantities. The resulting Euler–Lagrange equation defines the Master Entropic Equation (MEE), the universal field equation of ToE.

This chapter develops the mathematical foundations of ToE by integrating recent advances in information geometry—particularly the Riemannian interpretation of the Amari–Čencov α‑connections—into the structure of the Obidi Action. These results provide a rigorous geometric and variational framework for the Local Obidi Action (LOA) and the Spectral Obidi Action (SOA), establishing ToE as a unified entropic field theory.

4.2 Information Geometry as the Natural Language of Entropic Fields

Information geometry studies spaces of probability densities as differentiable manifolds equipped with canonical geometric structures. Two of these structures are central to ToE: the Fisher–Rao metric and the Amari–Čencov α‑connections.

Let ${\mathrm{D}\mathrm{e}\mathrm{n}\mathrm{s}}^{+}(M)$ denote the space of smooth positive densities on a manifold $M$. For $\rho \in {\mathrm{D}\mathrm{e}\mathrm{n}\mathrm{s}}^{+}(M)$ and tangent vectors $a,b\in T_{\rho }{\mathrm{D}\mathrm{e}\mathrm{n}\mathrm{s}}^{+}(M)$, the Fisher–Rao metric is defined by

$$G_{\mathrm{F}\mathrm{R}}(a,b)={\int }_M\frac{ab}{\rho }d\mu \mathrm{.}$$

The Amari–Čencov α‑connections are defined by

$${\mathrm{\nabla }}_a^{(\alpha )}b=Db[a]-\frac{1+\alpha }{2}\frac{ab}{\rho },$$

where $Db[a]$ denotes the directional derivative of $b$ along $a$.

These structures interpolate between the mixture connection ($\alpha =-1$), the exponential connection ($\alpha =1$), and the Levi‑Civita connection of the Fisher–Rao metric ($\alpha =0$). In classical information geometry, they describe statistical models and divergence structures. In ToE, they become the geometric infrastructure of the entropic field.

4.3 The Riemannian Interpretation of α‑Connections

A major mathematical development arises from the discovery that for every real $\alpha$, there exists a Riemannian metric $G_{\alpha }$ whose Levi‑Civita connection is exactly the Amari–Čencov α‑connection. This metric, the α‑Fisher–Rao metric, is defined by

$$G_{\alpha }(a,b)={\int }_M{\rho }^{-\alpha -1}abd\mu \mathrm{.}$$

This result has profound implications for ToE:

1. It establishes α‑connections as genuine geometric objects, not merely statistical constructs.

2. It provides a variational interpretation of α‑geodesics as energy‑minimizing curves.

3. It legitimizes the use of α‑connections in the Obidi Action as the geometric expression of entropic dynamics.

The Levi‑Civita connection of $G_{\alpha }$ is

$${\mathrm{\nabla }}_a^{(\alpha )}b=Db[a]-\frac{1+\alpha }{2}\frac{ab}{\rho },$$

precisely matching the Amari–Čencov α‑connection. This equivalence provides the geometric backbone for the LOA.

4.4 The Local Obidi Action (LOA) and Entropic Geodesics

The LOA is the geometric formulation of ToE, constructed from local entropic densities. Using the α‑Fisher–Rao metric, the LOA takes the form

$$S_{\mathrm{L}\mathrm{O}\mathrm{A}}[\rho ]={\int }_{\mathcal{M}}{G}_{\alpha }(\mathrm{\nabla }\rho ,\mathrm{\nabla }\rho )d\mu \mathrm{.}$$

The Euler–Lagrange equation of this action yields the α‑geodesic equation

$${\mathrm{\nabla }}_{\dot{\rho }}^{(\alpha )}\dot{\rho }=0.$$

Thus, the entropic field evolves along α‑geodesics, and the LOA becomes a Riemannian variational principle. This provides a rigorous geometric interpretation of the MEE.

4.5 The Spectral Obidi Action (SOA) and Araki Relative Entropy

The SOA lifts ToE into the operator‑algebraic domain. Let ${\mathrm{\Delta }}_{\rho \mathrm{\mid }\sigma }$ denote the relative modular operator associated with two states $\rho$ and $\sigma$. The Araki relative entropy is

$$S(\rho \mathrm{\parallel }\sigma )=-⟨\log {\mathrm{\Delta }}_{\rho \mathrm{\mid }\sigma }{⟩}_{\rho }\mathrm{.}$$

ToE promotes this quantity from a diagnostic tool to a generative principle by defining

$$S_{\mathrm{S}\mathrm{O}\mathrm{A}}[\rho ]={\int }_{\mathcal{M}}S(\rho (x)\mathrm{\parallel }\sigma (x))d\mu ,$$

where $\sigma$ is a reference entropic configuration.

The α‑geometry results justify this construction by showing that entropic divergences naturally generate geometric connections and geodesics. The SOA thus becomes the spectral counterpart of the LOA, and the two are related by a duality analogous to the α ↔ −α duality of information geometry.

4.6 Flattening Entropic Geometry: α‑Root Maps and Linearization

A remarkable result from the α‑geometry literature is that the α‑Fisher–Rao metric becomes flat under the α‑root map

$${\mathrm{\Phi }}_{\alpha }(\rho )=\{\begin{array}{cc}\frac{2}{1-\alpha }{\rho }^{\frac{1-\alpha }{2}},& \alpha \mathrm{\ne }1,\\ \log \rho ,& \alpha =1.\end{array}$$

Under this transformation, α‑geodesics become straight lines in function space. This flattening mechanism parallels the spectral flattening induced by modular operators in the SOA. It demonstrates that entropy possesses a natural linearizing transformation, supporting the idea that the entropic field has both geometric and spectral representations.

4.7 Hydrodynamic PDEs as Entropic Geodesics

One of the most striking results from the attached paper is the connection between α‑geodesics and hydrodynamic PDEs. When α‑connections on density spaces are pulled back to diffeomorphism groups, they yield nonlinear PDEs such as the generalized Proudman–Johnson equation

$$u_{txx}+(2-\alpha )u_x{u}_{xx}+u{u}_{xxx}=0.$$

This provides a concrete example of the ToE principle that physical dynamics emerge from entropic geometry. It suggests that Einstein’s equations, Yang–Mills dynamics, and quantum evolution may similarly arise from the MEE.

4.8 Entropic Duality and the Structure of Physical Law

The α‑connections satisfy the duality relation

$${\mathrm{\nabla }}^{(\alpha )}\text{ is dual to }{\mathrm{\nabla }}^{(-\alpha )}\text{ under }{G}_{\mathrm{F}\mathrm{R}}\mathrm{.}$$

This duality mirrors the LOA–SOA duality in ToE:

LOA corresponds to local geometric evolution. SOA corresponds to spectral evolution under modular operators.

The α‑geometry results provide a rigorous mathematical foundation for this duality, suggesting that physical law may arise from the interplay between geometric and spectral entropic structures.

4.9 Conclusion: Toward a Unified Entropic Field Theory

The integration of α‑geometry into the Theory of Entropicity yields a coherent and mathematically rigorous framework in which entropy is the fundamental field of the universe. The α‑Fisher–Rao metrics, α‑connections, and their Riemannian interpretation provide the geometric structure needed for the LOA. The spectral properties of Araki relative entropy provide the operator‑theoretic structure needed for the SOA. The emergence of hydrodynamic PDEs from entropic geodesics demonstrates the physical generativity of the framework.

ToE thus stands as a unified entropic field theory in which geometry, quantum mechanics, thermodynamics, and physical dynamics all arise from the intrinsic structure and evolution of entropy itself.

References

1. Bauer, Martin, Alice Le Brigant, and Cy Maor. “A Riemannian Viewpoint on the Amari–Čencov α‑Connections and Proudman–Johnson Equations.” arXiv (1 August, 2025). https://arxiv.org/abs/2508.00371

2. https://theoryofentropicity.blogspot.com/2026/01/what-are-applications-and-differences.html

3. https://theoryofentropicity.blogspot.com/2026/01/on-john-onimisi-obidis-ingenious.html

4. A Simplistic Expansion and Solution of a Trivial Form of the Obidi Field Equations (OFE) of the Theory of Entropicity (ToE): https://theoryofentropicity.blogspot.com/2026/01/a-simplistic-expansion-and-solution-of.html

Theory of Entropicity (ToE) — The Yuletide Papers

The Yuletide Papers

1. Obidi, John Onimisi. Collected Works on the Evolution of the Foundations of the Theory of Entropicity(ToE): Establishing Entropy as the Fundamental Field that Underlies and Governs All Observations, Measurements, and Interactions - Volume I: The Conceptual and Philosophical Expositions (Version 1.0). (December 31, 2025). The Yuletide Volume. Available at SSRN: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5996415
2. Obidi, John Onimisi (30th December, 2025). From the Temperature of Information to the Temperature of Geometry: The Foundations of the Theory of Entropicity (ToE) and the Unification of Quantum and Entropic Reality - A Unified Framework for the Thermodynamic, Quantum, and Geometric Foundations of Physical Reality. Figshare. https://doi.org/10.6084/m9.figshare.30976342
3. Obidi, John Onimisi (29th December, 2025). The Theory of Entropicity (ToE) Sheds Light on String Theory, Quantum Field Theory, and the Casimir Effect: Strings and Branes are Vibrations of Information [Geometry] in the Entropic Field of ToE. Figshare. https://doi.org/10.6084/m9.figshare.30968344
4. Obidi, John Onimisi (28th December, 2025). Entropicity, Neutrino Mixing, and the PMNS Matrix:  A New Perspective on Neutrino Oscillations and Symmetries Based on New Insights from the Theory of Entropicity(ToE). Figshare. https://doi.org/10.6084/m9.figshare.30964483
5. Obidi, John Onimisi (28th December, 2025). Further Expositions on the Theory of Entropicity (ToE) and Ginestra Bianconi’s Gravity from Entropy: How the Theory of Entropicity (ToE) Unifies Spectral and Araki Entropies with Tsallis, Rényi, Fisher–Rao, Fubini–Study, and Amari–Čencov Formalisms. Figshare. https://doi.org/10.6084/m9.figshare.30959819
6. Obidi, John Onimisi (27th December 2025). The Theory of Entropicity (ToE) Goes Beyond Holographic Pseudo-Entropy: From Boundary Diagnostics to a Universal Entropic Field Theory. Figshare. https://doi.org/10.6084/m9.figshare.30958670