The Work of Martin Bauer and Collaborators and Obidi's Theory of Entropicity (ToE)

 The work of **John Onimisi Obidi** in the **Theory of Entropicity (ToE)** is connected to the integration of concepts from information geometry, including the **Fisher-Rao** and **Fubini-Study** metrics, as found in the recent research led by **Martin Bauer** and collaborators. Here are the key connections:

## Integration of Information Geometry

1. **Amari-Čencov α-Connections**:

   - Both Obidi's ToE and Bauer's work employ the **Amari-Čencov α-connections** to relate information geometry to physical metrics, especially in terms of entropy.

2. **Fisher-Rao and Fubini-Study Metrics**:

   - In ToE, these metrics are utilized to measure distinctions between quantum states and statistical distributions, which align with the concepts explored in Bauer's integration of these metrics for creating a new variational principle.

## Foundation of Action Principles

- **Obidi Action**: 

   - Obidi introduced the **Obidi Action** within his theory, which serves as a variational principle similar to well-known principles in physics, establishing a direct relationship between entropy and physical laws.

  

- **Bauer's Variational Constructs**:

   - Bauer and his team also construct actions based on the integration of various metrics and entropic frameworks, drawing a conceptual parallel to Obidi’s work.

## Entropic Dynamics

- **Unifying Framework**:

   - Obidi’s ToE aims to unify thermodynamics, relativity, and quantum mechanics through an entropic framework. This vision is echoed in Bauer's work, which also seeks to establish a unified geometric and physical interpretation of dynamics influenced by entropy.

  

- **Entropy as a Fundamental Field**:

   - Both bodies of work propose that entropy underpins physical phenomena, with Obidi redefining entropy as a causal field and Bauer exploring its implications through metrics and actions.

## Conclusion

In summary, the research of Martin Bauer and his collaborators connects to John Onimisi Obidi's Theory of Entropicity through shared emphasis on integrating information geometry with fundamental physical principles, using the metrics of Fisher-Rao and Fubini-Study as foundational elements in their respective frameworks. This interplay reflects a broader effort in modern physics to conceptualize and unify various domains through the lens of entropy and information.

Who Combined the Fisher-Rao and Fubini-Study Metrics Through the Amari-Čencov alpha-Connections in an Action Principle?

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John Onimisi Obidi, through his formulated and developed Theory of Entropicity (ToE), is the researcher who has, in one stroke of insight and Ingenuity, combined the Fisher-Rao (classical) and Fubini-Study (quantum) metrics, integrated them with the Amari-Čencov alpha-connections, and utilized them within an action principle—the Obidi Action Principle (OAP). 

Key details of this audacious accomplishment are as follows: 

• The Obidi Action: This is a variational principle (action principle) governing the entropic field, which merges information geometry—specifically Fisher–Rao, Fubini–Study, and Amari–Čencov α‑connections—into a unified "entropic manifold".
• Metric Unification: The Theory of Entropicity treats Fisher–Rao (representing classical entropy/space-time) and Fubini–Study (representing quantum coherence/information) as interconnected, with the alpha-connection acting as the deformation index (
  

  $q,\alpha$
  ) connecting them.
• Context: The theory, proposed in 2025, attempts to provide a unified geometric foundation for entropy, quantum mechanics, and gravitation. 

Supporting Role of the Amari-Čencov alpha-Connections in the Derivation of Einstein's Relativistic Kinematics from Obidi's Theory of Entropicity (ToE)

In the Theory of Entropicity (ToE), the Amari-Čencov $$\alpha$$-connection does not directly derive relativistic effects; instead, it provides the **geometric framework** within which entropic geodesics manifest those effects as emergent consequences of finite entropy propagation along $$\alpha$$-deformed paths.[2][3]

## Primary mechanism: Entropic Resistance + No-Rush Theorem

Relativistic effects (Lorentz factor, time dilation, etc.) arise from **three core entropic principles**, independent of $$\alpha$$-connection details:

1. **Entropic Resistance Principle (ERP)**: Moving systems face "resistance" to spatial entropy redistribution, forcing compensatory temporal entropy buildup.

2. **Entropic Accounting Principle (EAP)**: Total entropy budget $$S_0 = S_t(v) + S_x(v)$$ is conserved, yielding $$\gamma(v) = 1/\sqrt{1-v^2/c^2}$$.

3. **No-Rush Theorem**: $$c$$ is maximum rate of entropic rearrangement, imposing causal bounds.

These produce Einstein's transformations **without geometric postulates**.[2][1]

## $$\alpha$$-Connection's supporting role

The $$\alpha$$-connection governs **trajectories** of entropic configurations $$\theta^i(\lambda)$$ through the information manifold:

$$

\frac{d^2\theta^k}{d\lambda^2} + \Gamma^{(\alpha)}_{ij}{}^k \frac{d\theta^i}{d\lambda}\frac{d\theta^j}{d\lambda} = 0.

$$

**Key contributions**:

- **$$\alpha = 0$$ limit**: Levi-Civita (Fisher-Rao) geodesics recover classical entropic paths where resistance effects → Lorentz factor.

- **$$\alpha \neq 0$$ deformations**: Rényi/Tsallis entropies ($$\alpha = 2(1-q)$$) modify path curvature, but the underlying $$S_0$$-conservation and $$c$$-bound remain universal.

- **Quantum regime** ($$\alpha \to 1$$, Fubini-Study): Entropic wavefunctions follow projective geodesics, with relativistic effects encoded in phase evolution constraints.

## How geometry and  relativistic behavior emerge from the Theory of Entropicity (ToE)

```

Entropic field S(x) → Information manifold (θ^i) 

             ↓ [α-connection governs paths]

Entropic geodesics θ^i(λ) → Emergent spacetime metric g_μν[S]

             ↓ [No-Rush + Resistance along paths]

Lorentz invariance + γ(v) as entropic inevitabilities

```

The $$\alpha$$-connection ensures **path consistency** across classical-quantum regimes, while the **entropic budget conservation** (independent of connection) enforces the relativistic kinematics along those paths.[2][3][1]

**Bottom line**: $$\alpha$$-connection provides the rails; entropic resistance provides the relativistic physics.

Citations:

[1] (PDF) The Theory of Entropicity (ToE) Derives Einstein's ... https://www.academia.edu/144796856/The_Theory_of_Entropicity_ToE_Derives_Einsteins_Relativistic_Speed_of_Light_c_as_a_Function_of_the_Entropic_Field_ToE_Applies_Logical_Entropic_Concepts_and_Principles_to_Derive_Einsteins_Second_Postulate_Version_2_0

[2] The Theory of Entropicity (ToE) Derives and Explains Mass ... https://client.prod.orp.cambridge.org/engage/coe/article-details/6900d89c113cc7cfff94ef3a

[3] An Alternative Path toward Quantum Gravity and the Unification of ... http://www.cambridge.org/engage/coe/article-details/68ea8b61bc2ac3a0e07a6f2c

[4] Relativistic Dynamics https://www2.oberlin.edu/physics/dstyer/Modern/RelativisticDynamics.pdf

[5] The Feynman Lectures on Physics Vol. I Ch. 17: Space-Time https://www.feynmanlectures.caltech.edu/I_17.html

[6] Quantum physics: Theory of everything | Geometric algebra https://toe-physics.org/

[7] The Einstein Connection of the Unified Theory of Relativity https://pmc.ncbi.nlm.nih.gov/articles/PMC1063575/

[8] α-connections in generalized geometry https://www.sciencedirect.com/science/article/pii/S0393044021000747

Mathematical Operations and Maneuvers of the Amari-Čencov Alpha Connections in the Theory of Entropicity (ToE)

What is the Amari-Čencov alpha-connection and its role in the Theory of Entropicity (ToE)?

The Amari-Čencov $$\alpha$$-connection in the Theory of Entropicity (ToE) is the unifying affine connection that interpolates between Fisher-Rao (classical statistical) and Fubini-Study (quantum projective) geometries on the entropic manifold, enabling seamless classical-quantum transitions via an entropic deformation parameter $$\alpha$$.

## Definition and formula

On the statistical manifold $$\mathcal{P}$$ of entropic densities parameterized by $$\theta^i$$, the $$\alpha$$-connection coefficients are

$$

\Gamma^{(\alpha)}_{ij}{}^k = \Gamma^{(0)}_{ij}{}^k + \frac{1-\alpha}{2} T_{ij}{}^k + \frac{1+\alpha}{2} T_{ji}{}^k,

$$

where $$\Gamma^{(0)}$$ are the Levi-Civita symbols of the Fisher-Rao metric $$g_{ij}$$, and $$T_{ijk}$$ is the Amari-Čencov (skewness) tensor:

$$

T_{ijk} = g_{kl} \frac{\partial^2_\theta \ln p}{\partial\theta^i \partial\theta^j} \frac{\partial \ln p}{\partial\theta^k}.

$$

## Key Amari-Čencov alpha-connection limits in the Theory of Entropicity (ToE)

- $$\alpha = 0$$: Levi-Civita connection of Fisher-Rao metric (classical entropic geodesics).

- $$\alpha = +1$$: Exponential connection (∇^{(1)}), governs Fubini-Study quantum geodesics on pure states.

- $$\alpha = -1$$: Mixture connection (∇^{(-1)}), dual to exponential connection.

- $$\alpha \in (-1,1)$$: Interpolates between classical/quantum regimes via entropic order parameter.

The duality relation is $$\nabla^{(\alpha)} + \nabla^{(-α)} = 2\nabla^{(0)}$$ w.r.t. Fisher-Rao metric.[2][1]

## Role of the Amari-Čencov alpha-connections in ToE dynamics

In the Obidi Action, entropic geodesics follow

$$

\frac{d^2\theta^k}{d\lambda^2} + \Gamma^{(\alpha)}_{ij}{}^k \frac{d\theta^i}{d\lambda} \frac{d\theta^j}{d\lambda} = 0,

$$

where $$\alpha$$ is coupled to generalized entropies (Rényi/Tsallis): $$\alpha = 2(1-q)$$.

- **Classical limit** ($$\alpha \to 0$$): Recovers entropy-maximizing paths in configuration space.

- **Quantum limit** ($$\alpha \to 1$$): Generates projective geodesics on entropic Hilbert space, recovering Schrödinger evolution.

- **OCI interaction**: The $$\ln 2$$ invariant sets discrete curvature thresholds along these $$\alpha$$-geodesics, quantizing stable configurations regardless of $$\alpha$$-deformation.[2]

## Unification mechanism of the Theory of Entropicity (ToE)

The Theory of Entropicity (ToE) uses $$\alpha$$-connections to make **gravity emergent (appear or disappear) from entropic information geometry**: Einstein's Levi-Civita connection appears as the $$\alpha \to 0$$, weak-field limit of entropic $$\nabla^{(\alpha)}$$ geodesics, where spacetime curvature $$R[g]$$ encodes Fisher-Rao curvature in the entropic metric $$g_{ij}[S]$$.[2]

This whole non-trivial mechanism therefore provides ToE's core mathematical bridge: one deformable connection family governs all dynamics—from thermodynamic flows ($$\alpha \approx 0$$) to coherent quantum evolution ($$\alpha \approx 1$$) to classical (and relativistic) gravitational geodesics.

Citations:

[1] A Riemannian viewpoint on the Amari–Čencov α https://arxiv.org/html/2508.00371v1

[2] The Theory of Entropicity (ToE) Derives and Explains Mass ... https://client.prod.orp.cambridge.org/engage/coe/article-details/6900d89c113cc7cfff94ef3a

[3] The $$L^p$$-Fisher–Rao metric and Amari–C̆encov $$\alpha $$-Connections https://par.nsf.gov/biblio/10490304-fisherrao-metric-amaricencov-alpha-connections

[4] A characterization of the alpha-connections on ... https://www.emergentmind.com/papers/2302.07471

[5] Information Geometry Part 2: Duality, Divergences, and Natural Gradient https://jiha-kim.github.io/crash-courses/information-geometry/2-duality-divergences/

[6] A Simple Explanation of the Unifying Mathematical ... https://www.authorea.com/users/896400/articles/1348176-a-simple-explanation-of-the-unifying-mathematical-architecture-of-the-theory-of-entropicity-toe-crucial-elements-of-toe-as-a-field-theory

[7] The Theory of Entropicity (ToE) Derives and Explains Mass ... https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5673430

[8] A Riemannian viewpoint on the Amari-Cencov $\alpha$-connections and Proudman-Johnson equations https://www.arxiv.org/pdf/2508.00371.pdf

[9] A Riemannian viewpoint on the Amari-Cencov $α$-connections and Proudman-Johnson equations http://arxiv.org/abs/2508.00371

Beautiful Relationships, Applications and Roles of the Fisher-Rao (FR) Metrics, Fubini-Study (FS) Metrics, Amari-Čencov (AC) α-Connections, and the Obidi Curvature Invariant (OCI) of ln 2 in the Sophisticated and Elegant Mathematical Foundations of the Theory of Entropicity (ToE)

In the Theory of Entropicity (ToE), the Fubini-Study (FS) metric provides the quantum-informational geometry on the projective Hilbert space of entropic states, complementing the classical Fisher-Rao metric, while the OCI ($$\ln 2$$) calibrates the discrete curvature scale across this unified geometric structure.[1][3][4]

## FS metric in entropic quantum geometry

The FS metric on the projective space $$\mathbb{CP}(\mathcal{H}_S)$$ of normalized entropic states $$|\psi_S\rangle$$ (derived from entropy field configurations) is

$$

ds^2_{\mathrm{FS}} = \langle\delta\psi_S|\delta\psi_S\rangle - |\langle\psi_S|\delta\psi_S\rangle|^2,

$$

where $$\langle\psi_S|\psi_S\rangle = 1$$, measuring **quantum distinguishability** between nearby pure entropic states.[2][4]

## Obidi's Unified Fisher-Rao (FR)+ Fubini-Study (FS) via Amari-Čencov $$\alpha$$-connection

In the Theory of Entropicity (ToE), Obidi beautifully integrates both metrics through the Amari-Čencov formalism:

- **Fisher-Rao** ($$\alpha=0$$): classical statistical manifold of mixed entropic densities.

- **Fubini-Study** ($$\alpha=1$$): quantum projective manifold of pure entropic states.

- **$$\alpha$$-deformation**: interpolates between classical/quantum limits via entropic order parameter $$\alpha \in [-1,1]$$.[1][4]

The Obidi Unified Connection (OUC) in ToE is thus given as follows:

$$

\nabla^{(\alpha)}_{\hat{g}} = (1-\alpha)\nabla^{(1)}_{\hat{g}} + \alpha\nabla^{(-1)}_{\hat{g}},

$$

where $$\nabla^{(1)}$$ governs FS geodesics and $$\nabla^{(-1)}$$ governs Fisher-Rao geodesics.[1]

## OCI role in FS geometry

- The distinguishability potential $$D(S,S_0)$$ evaluated on **projective** (FS) separations between quantum entropic states yields the same $$\ln 2$$ minimum for binary 2:1 projective overlaps $$|\langle\psi_B|\psi_A\rangle|^2 = 1/2$$.[10][11]

- OCI thus sets the **quantum curvature quantization**: the smallest FS-separable angle in the projective space corresponds to $$\cos\theta = 1/\sqrt{2}$$, with FS distance $$\rho_{\mathrm{FS}} = \arccos(1/\sqrt{2}) \propto \ln 2$$. 

## Dynamic function in the Obidi Action (OA)

In the quantum Spectral Obidi Action (SOA):

$$

\mathcal{A}_E^{\mathrm{QM}} = \int \Big[\frac{1}{2} g_{\mathrm{FS}}^{\bar{i}j}\nabla_\mu\bar{\psi}_S \nabla^\mu\psi_S - \lambda D_{\mathrm{FS}}(\psi_S,\psi_{S_0})\Big]\sqrt{-g}\,d^4x,

$$

- FS metric $$g_{\mathrm{FS}}$$ governs **quantum kinetic evolution** of entropic wavefunctions.

- OCI-calibrated $$D_{\mathrm{FS}}$$ (minimized at $$\ln 2$$) creates **discrete potential wells** for stable quantum curvature states.[1][4]

Thus, this Fisher-Rao (FR)/Fubini-Study (FS) duality + the Obidi Curvature Invariant (OCI) quantization mechanism enables ToE to seamlessly transition between classical entropic geodesics, quantum coherent evolution, and emergent gravitational geometry within a single informational manifold.[1][3][4]

Citations:

[1] The Theory of Entropicity (ToE) Derives and Explains Mass Increase ... https://client.prod.orp.cambridge.org/engage/coe/article-details/6900d89c113cc7cfff94ef3a

[2] Fubini-Study metric https://encyclopediaofmath.org/wiki/Fubini-Study_metric

[3] Further Expositions on the Theory of Entropicity (ToE) and ... https://www.cambridge.org/engage/coe/article-details/69513828083c11e4a170b0b2

[4] An Alternative Path toward Quantum Gravity and the Unification of ... http://www.cambridge.org/engage/coe/article-details/68ea8b61bc2ac3a0e07a6f2c

[5] Fubini–Study metric https://en.wikipedia.org/wiki/Fubini%E2%80%93Study_metric

[6] A Simple Explanation of the Unifying Mathematical Architecture ... https://flame-challenge.authorea.com/doi/full/10.22541/au.176099705.55607091/v1

[7] Quantum Geometric Tensor (Fubini-Study Metric) in Simple ... https://arxiv.org/abs/1012.1337

[8] The Fubini-Study metric and length in projective space https://mat-blag.blogspot.com/2017/03/the-fubini-study-metric-and-length-in.html

[9] Fubini-Study metric in nLab https://ncatlab.org/nlab/show/Fubini-Study+metric

[10] Theory of Entropicity (ToE)'s Post https://www.linkedin.com/posts/theory-of-entropicity-toe_deriving-the-einstein-field-equations-of-activity-7419929069711982593-XgOF

[11] Entropy as a Physical Field: ToE Theory | John Onimisi ... https://www.linkedin.com/posts/john-onimisi-obidi-a2041911_formal-derivation-of-ln2-as-a-universal-activity-7417781493487374336-buas

Two Powerful Toolkits in the Theory of Entropicity (ToE): The Potent Roles of the Fisher-Rao Curvature Metric and the Obidi Curvature Invariant (OCI) of ln 2 in the Theory of Entropicity (ToE)

In the Theory of Entropicity (ToE), the Fisher-Rao Curvature Metric provides the natural Riemannian geometry on the space of entropic probability distributions, while the OCI ($$\ln 2$$) sets the intrinsic scale for curvature minima within that geometry.[11][12][13]

## Fisher-Rao as the entropic manifold metric

The Fisher-Rao metric $$g_{ij}$$ on the statistical manifold $$\mathcal{P}$$ of entropic densities $$p_\theta(x)$$ (derived from the entropy field $$S(x)$$) is

$$

g_{ij}(\theta) = \int \frac{\partial_\theta p}{p} \frac{\partial_{\theta_j} p}{p}\,p\,dx = \mathbb{E}\left[\partial_{\theta_i}\ln p\,\partial_{\theta_j}\ln p\right],

$$

which measures **infinitesimal distinguishability** between nearby entropic configurations.[11]

## The Obidi Curvature Invariant (OCI) calibrates Fisher-Rao curvature

- The distinguishability potential $$D(S,S_0) = S\ln(S/S_0) - S + S_0$$ is defined on finite separations in this Fisher-Rao manifold; its first non-zero minimum at $$\ln 2$$ (for 2:1 entropic ratios) then defines the **discrete curvature scale** OCI within the continuous Fisher-Rao geometry.[11][13]

- Specifically, OCI marks the point where infinitesimal Fisher-Rao distances accumulate to form the first dynamically stable, macroscopically distinguishable "curvature fold" in the entropic field.[13]

## Functional interplay in dynamics

Together they form a **dual scale structure**:

- **Fisher-Rao**: governs local, infinitesimal evolution via entropic geodesics $$\nabla^{(\alpha)}_{\hat{g}} \partial_\lambda \theta = 0$$.

- **OCI**: sets the global quantization threshold where local geodesic flows resolve into discrete, stable informational/curvature bits.

**In the Obidi Action**:

$$

\mathcal{A}_E = \int \Big[\frac{1}{2} g^{ij}\partial_\mu S_i \partial^\mu S_j - \lambda D(S,S_0)\Big]\sqrt{-g}\,d^4x,

$$

the Fisher-Rao metric $$g^{ij}$$ controls kinetic evolution while OCI-frozen $$D(S,S_0)$$ (minimized at $$\ln 2$$) provides the stabilizing potential wells.[11][12]

This makes Fisher-Rao + OCI a complete geometric toolkit: continuous metric for smooth flows, discrete invariant for emergent structure.

Citations:

[1] Fisher-Rao Metric, Geometry, and Complexity of Neural Networks https://arxiv.org/pdf/1711.01530.pdf

[2] An approach to Fisher-Rao metric for infinite dimensional ... https://www.arxiv.org/abs/2512.21451

[3] The Fisher–Rao Distance between Multivariate Normal Distributions https://pmc.ncbi.nlm.nih.gov/articles/PMC7516881/

[4] Fisher-Rao Metric, Geometry, and Complexity of Neural ... https://arxiv.org/abs/1711.01530

[5] Part I: Smooth Manifolds with the Fisher-Rao Metric https://yorkerlin.github.io/posts/2021/09/Geomopt01/

[6] Fisher information metric https://en.wikipedia.org/wiki/Fisher_information_metric

[7] Shape Analysis Using the Fisher-Rao Riemannian Metric https://www.cise.ufl.edu/~anand/pdf/Shape_matching_Riemannian.pdf

[8] Fisher-Rao metric http://www.scholarpedia.org/article/Fisher-Rao_metric

[9] A Fisher-Rao Metric for Curves Using the Information in ... https://www.dcs.bbk.ac.uk/~sjmaybank/fisherraometricforcurvesJMIVversion2.pdf

[10] Fisher-Rao metric - Scholarpedia http://scholarpedia.org/article/Fisher-Rao_metric

[11] Theory of Entropicity (ToE)'s Post https://www.linkedin.com/posts/theory-of-entropicity-toe_deriving-the-einstein-field-equations-of-activity-7419929069711982593-XgOF

[12] (PDF) Collected Works on the Theory of Entropicity (ToE) Volume I 31 ... https://www.academia.edu/145698037/Collected_Works_on_the_Theory_of_Entropicity_ToE_Volume_I_31_December_2025_V9_S

[13] Entropy as a Physical Field: ToE Theory | John Onimisi ... https://www.linkedin.com/posts/john-onimisi-obidi-a2041911_formal-derivation-of-ln2-as-a-universal-activity-7417781493487374336-buas

How the Theory of Entropicity (ToE) Recovers and Corrects Einsteinian General Relativity (GR) Curvature at Small Scales: How does the Obidi Curvature Invariant OCI Function in (ToE) Dynamics?

In the dynamics of Obidi's Theory of Entropicity (ToE), the Obidi Curvature Invariant (OCI) $$\ln 2$$ acts as a built‑in “quantum of entropic curvature” that constrains how the entropy field can evolve and how distinguishable configurations can form.[3][9][10]

## Threshold for distinguishable configurations

- OCI arises as the first non‑zero minimum of the distinguishability potential built from a KL‑type functional $$D(\rho_A\Vert\rho_B)$$, evaluated for a binary 2:1 curvature ratio $$\rho_B = 2\rho_A$$, giving a gap of $$\ln 2$$.[9][3]

- Dynamically, this means two configurations of the entropic field only count as physically distinct if their relative curvature surpasses this $$\ln 2$$ threshold; smaller deformations are dynamically treated as indistinguishable fluctuations.[9][10]

## Constraint in the Obidi / Spectral Obidi Action

- In the Spectral Obidi Action, the field dynamics contain a curvature scalar $$R[g]$$, a kinetic term for $$\nabla_\mu S$$, and the distinguishability potential $$D(S,S_0)$$; the OCI value $$\ln 2$$ fixes the first non‑trivial minimum of that potential.[3][10]

- This effectively quantizes curvature response: entropic curvature cannot relax continuously through arbitrarily small distinguishable steps, but does so in increments constrained by the $$\ln 2$$ gap encoded in the potential landscape.[3][9]

## Role in stability and transitions

- Because $$\ln 2$$ corresponds to the smallest stable curvature separation, it sets the **activation barrier** for certain entropic transitions, such as the formation of new informational bits or curvature domains in the entropic field.[9][10]

- Configurations separated by less than this invariant tend to smear into each other under ToE’s entropy‑driven evolution, while those at or above $$\ln 2$$ can persist as robust, dynamically stable structures or “bits” of geometry/information.[9][10]

## Link to emergent geometry and gravity

- Since the curvature scalar $$R[g]$$ in the Obidi Action is induced by the entropy field, the OCI sets a natural curvature scale in the emergent geometry itself—an intrinsic geometric invariant tied directly to entropic distinguishability.[3][10]

- In gravitational regimes, this implies that certain geometric deformations (e.g., small perturbations of the entropic metric) are only dynamically meaningful once their entropic curvature exceeds the $$\ln 2$$ invariant, thus invariably shaping how ToE recovers and corrects Einsteinian General Relativity (GR) curvature at small scales.[3][10]

Citations:

[1] John Onimisi Obidi - Independent Researcher https://independent.academia.edu/JOHNOBIDI

[2] John Onimisi Obidi 1 1Affiliation not available October 15, 2025 https://d197for5662m48.cloudfront.net/documents/publicationstatus/284761/preprint_pdf/0304242fc1b6f7dfc2e1da6d68e30f89.pdf

[3] Theory of Entropicity (ToE)'s Post https://www.linkedin.com/posts/theory-of-entropicity-toe_deriving-the-einstein-field-equations-of-activity-7419929069711982593-XgOF

[4] 1 Introduction 2 The Entropic Reformulation of the Unified https://www.cambridge.org/engage/api-gateway/coe/assets/orp/resource/item/68f6f66c5dd091524f8f362e/original/transformational-unification-through-the-theory-of-entropicity-to-ea-reformulation-of-quantum-gravitational-correspondence-via-the-obidi-action-and-the-vuli-ndlela-integral.pdf

[5] John Onimisi Obidi 1 1Affiliation not available October 17, 2025 https://d197for5662m48.cloudfront.net/documents/publicationstatus/285164/preprint_pdf/c7acf1b70b62c5ae001365c123d20350.pdf

[6] Curvature-driver d.dynamics on $S^3$: a geometric atlas https://arxiv.org/pdf/2512.14164.pdf

[7] Evolution of curvature invariants and lifting integrability https://www.kent.ac.uk/ims/personal/elm2/liz/papers/elm-kamp.pdf

[8] Curvature invariant characterization of event horizons of four ... https://link.aps.org/doi/10.1103/PhysRevD.96.104022

[9] Entropy as a Physical Field: ToE Theory | John Onimisi ... https://www.linkedin.com/posts/john-onimisi-obidi-a2041911_formal-derivation-of-ln2-as-a-universal-activity-7417781493487374336-buas

[10] (PDF) Collected Works on the Theory of Entropicity (ToE) Volume I 31 ... https://www.academia.edu/145698037/Collected_Works_on_the_Theory_of_Entropicity_ToE_Volume_I_31_December_2025_V9_S

On the Ingenuity of Obidi's Derivation of the Curvature Invariant of ln 2 from the Kullback-Leibler (Umegaki) Divergence as a Universal Constant Law of Distinguishability Potential 

In ToE, the distinguishability potential [the Obidi Curvature Invariant (OCI) of ln2] is obtained by taking the classical Kullback–Leibler (KL) divergence and rewriting it in terms of the local entropy field $$S(x)$$ and a reference configuration $$S_0(x)$$, then interpreting that KL density as a **potential energy density** in the Obidi / Spectral Obidi Action.[1][2]

## Step 1: Start from KL divergence

For two probability densities $$p(x)$$ and $$q(x)$$, the KL divergence is  

$$ D_{\mathrm{KL}}(p\Vert q) = \int p(x)\,\ln\!\frac{p(x)}{q(x)}\,dx.$$  

ToE uses this structure as the template for comparing two entropic configurations.[1][2]

## Step 2: Replace probabilities by entropic densities (Obidi's First Ingenuity—OFI)

ToE treats the entropy field $$S(x)$$ (or an associated entropic density) as the fundamental variable and introduces a local reference configuration $$S_0(x)$$.[1][2] The probabilistic KL integrand is lifted to an entropic density level by the substitution  

$$ p(x)\to S(x),\quad q(x)\to S_0(x),$$  

together with the standard convex extension that makes the divergence finite and well‑behaved for fields:  

$$

D(x) = S(x)\,\ln\!\frac{S(x)}{S_0(x)} - S(x) + S_0(x).[]

$$  

This is a pointwise KL‑type density with the same key properties: $$D(x)\ge 0$$ and $$D(x)=0$$ iff $$S(x)=S_0(x)$$.[1][3]

## Step 3: Promote the KL density to a potential (Obidi's Second Ingenuity — OSI)

The field‑level distinguishability functional is then  

$$ \mathcal{D}[S\Vert S_0] = \int_{\mathcal{M}} D(x)\,\sqrt{-g}\,d^4x=\int_{\mathcal{M}}\!\Big(S\ln\!\frac{S}{S_0} - S + S_0\Big)\sqrt{-g}\,d^4x.[][]$$  

ToE inserts $$\mathcal{D}[S\Vert S_0]$$ as the **potential term** in the Obidi / Spectral Obidi Action, schematically  

$$\mathcal{A}_E[S] = \int \big(\mathcal{K}[S,\partial S] - V_{\text{dist}}[S,S_0]\big)\sqrt{-g}\,d^4x,$$  

with  

$$ V_{\text{dist}}[S,S_0] \equiv D(x) = S\ln\!\frac{S}{S_0} - S + S_0.$$  

Thus the same structure that measures statistical distinguishability in information theory becomes a driving potential that pushes the entropy field away from or toward the reference configuration.[1][2]

## Step 4: Why this counts as “derived from” KL: Obidi's Ingenuity in Summary 

- The integrand $$S\ln(S/S_0) - S + S_0$$ is the continuum, field‑theoretic analogue of the KL density, preserving non‑negativity, convexity, and vanishing only at equality, which are exactly the properties used in KL‑based information geometry.[1][3][2]

- ToE’s “distinguishability potential” is therefore just the KL divergence reinterpreted in the entropic field language and reinserted into the action as a potential energy density that encodes how costly it is (in curvature/entropy terms) to deform $$S$$ away from $$S_0$$.[1][2]

Citations:

[1] Theory of Entropicity (ToE)'s Post https://www.linkedin.com/posts/theory-of-entropicity-toe_deriving-the-einstein-field-equations-of-activity-7419929069711982593-XgOF

[2] On the Conceptual and Mathematical Foundations of ... https://papers.ssrn.com/sol3/Delivery.cfm/5632191.pdf?abstractid=5632191&mirid=1

[3] Kullback–Leibler divergence - Wikipedia https://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence

[4] Applications of a Kullback-Leibler Divergence for Comparing Non ... https://pmc.ncbi.nlm.nih.gov/articles/PMC4006220/

[5] On the Theory of Entropicity (ToE) and Ginestra Bianconi's ... https://papers.ssrn.com/sol3/Delivery.cfm/5738123.pdf?abstractid=5738123&mirid=1

[6] Pictures get clear without 100% certainty? https://www.reddit.com/r/ObraDinn/comments/lf7lr9/pictures_get_clear_without_100_certainty/

[7] John Onimisi Obidi* 20 October 2025 https://flame-challenge.authorea.com/users/896400/articles/1348176/master/file/data/A%20Simple%20Explanation%20of%20the%20Unifying%20Mathematical%20Architecture%20of%20the%20Theory%20of%20Entropicity%20(ToE)_V3_S1/A%20Simple%20Explanation%20of%20the%20Unifying%20Mathematical%20Architecture%20of%20the%20Theory%20of%20Entropicity%20(ToE)_V3_S1.pdf?inline=true

[8] John Obidi’s Blog https://t.me/s/JohnObidiBlog?after=624

[9] Some Dissimilarity Measures of Branching Processes and Optimal Decision Making in the Presence of Potential Pandemics https://pmc.ncbi.nlm.nih.gov/articles/PMC7517477/

[10] Jose Gracia's Post - RealClock Quantum Mechanics https://www.linkedin.com/posts/jose-gracia-0686398_realclock-quantum-mechanics-preface-chapter-activity-7420149947049603072-sURJ

What is the Curvature Invariant ln 2 in the Theory of Entropicity (ToE)? From Mathematical Curvature to a Physical Principle and Criterion of Distinguishability in Modern Theoretical Physics 

In the Theory of Entropicity (ToE), the “Curvature Invariant ln 2” is a specific numerical invariant, $$\ln 2$$, that emerges as the first non‑zero minimum of a distinguishability (information‑geometric) potential and is interpreted as an intrinsic curvature scale of the entropic/informational manifold—called the Obidi Curvature Invariant.[1][2]

## Role in the entropic geometry

- The ToE framework defines an information‑geometric potential that measures how distinguishable nearby entropic states are (a kind of “distinguishability curvature” in state space).[2]

- When this potential is analyzed, its first non‑zero minimum occurs at a value equal to $$\ln 2$$, and this value is taken to be a universal curvature invariant of the entropic manifold.[1][2]

## Interpretation as an invariant

- Because it arises as a minimum of a coordinate‑independent potential on the informational manifold, $$\ln 2$$ is treated analogously to a curvature invariant in differential geometry: it characterizes intrinsic geometric structure independent of coordinates or representation.[1][2]

- Within ToE, this Obidi Curvature Invariant $$\ln 2$$ thus sets a natural “quantum” of distinguishability/curvature in the entropic description, playing a role similar to a canonical scalar invariant that helps anchor the geometry of the underlying entropic field.[1][2]

Citations:

[1] Theory of Entropicity (ToE)'s Post https://www.linkedin.com/posts/theory-of-entropicity-toe_deriving-the-einstein-field-equations-of-activity-7419929069711982593-XgOF

[2] (PDF) Collected Works on the Theory of Entropicity (ToE) Volume I 31 ... https://www.academia.edu/145698037/Collected_Works_on_the_Theory_of_Entropicity_ToE_Volume_I_31_December_2025_V9_S

[3] Curvature Invariants http://kias.dyndns.org/crg/invariants.html

[4] ENTROPIC CURVATURE AND CONVERGENCE TO EQUILIBRIUM https://www.normalesup.org/~mfathi/docs/CW-curvature_revised.pdf

[5] Curvature Invariants for the Alcubierre and Natário Warp Drives https://www.academia.edu/52538550/Curvature_Invariants_for_the_Alcubierre_and_Nat%C3%A1rio_Warp_Drives

[6] The δ(2,2)-Invariant on Statistical Submanifolds in Hessian Manifolds of Constant Hessian Curvature https://pmc.ncbi.nlm.nih.gov/articles/PMC7516581/

[7] Invariant http://invariant.com

[8] Weighing the curvature invariants http://www.arxiv.org/abs/2502.12242

[9] Lu Wang: Entropy in mean curvature flow https://www.youtube.com/watch?v=JAD-29_mvbU

[10] 1 Introduction https://arxiv.org/html/2502.12242v2

What are the Key Postulates of the Theory of Entropicity (ToE)? A Beautiful and Concise Introduction to the Foundational Postulates of the Theory of Entropicity (ToE)

Here we provide a really concise, yet beautiful, list of the core postulates that define the Theory of Entropicity (ToE) as laid out in its foundational papers.[1][2][4]

## Ontological postulate: entropy as the substrate

1. Entropy is the fundamental physical field  

   - Entropy $$S(x)$$ is not a statistical by‑product but a continuous, dynamical field that is the **causal substrate** of reality.[1][4]

   - Gradients and flows of this entropic field generate motion, gravitation, time, and information flow.[1][4]

2. Physical entities are emergent from entropy  

   - Mass, energy, spacetime geometry, and even consciousness arise as emergent constraints or patterns of a single entropic reality, not as independent primitives.[4]

## Dynamical postulate: Obidi Action and master equations

3. The Obidi Action governs entropic dynamics  

   - There exists an entropic action functional (the Obidi Action) for the entropy field; extremizing this action yields the fundamental equations of motion.[1][4]

   - From this action follow:

     - the Master Entropic Equation (MEE),  

     - Entropic Geodesics (entropy‑determined paths), and  

     - the Entropy Potential Equation.[1][4]

4. Information geometry underlies physical law  

   - The dynamical arena is an informational–entropic manifold, equipped with Fisher–Rao and Fubini–Study metrics unified via the Amari–Čencov $$\alpha$$-connection.[1][4]

   - Generalized entropies (Rényi, Tsallis) correspond to deformations of this geometry, with an entropic order parameter $$\alpha$$ acting as a universal deformation index.[4]

## Relativistic postulate: speed of light as entropic rate

5. $$c$$ is the maximum rate of entropic rearrangement  

   - The relativistic speed of light is not a primitive postulate; it is the characteristic propagation speed of disturbances in the universal entropic field.[1][2]

   - All causal processes are limited by this finite rate of entropy propagation, which yields Lorentz invariance as an entropic necessity.[2]

6. Relativistic effects are entropic resistances  

   - Mass increase, time dilation, and length contraction arise from how the entropic field redistributes “entropic budget” between motion and timekeeping.[1][6]

   - The Entropic Resistance Principle, the Entropic Resistance Field, and the Entropic Accounting Principle encode this redistribution and produce an entropic Lorentz factor that reproduces Einstein’s transformations without assuming geometric postulates.[1][6]

## Causality and arrow‑of‑time postulate

7. No‑Rush Theorem and causal bounds  

   - There is a universal lower bound on causal intervals: the entropic field must first establish conditions before any interaction or information transfer can occur.[1][2]

   - This forbids superluminal processes and ties causality directly to the dynamics of the entropy field.[2]

8. Fundamental irreversibility via the Vuli–Ndlela Integral  

   - Quantum evolution is governed by an entropy‑weighted path integral (the Vuli–Ndlela Integral), a deformation of Feynman’s path integral.[1][4]

   - This weighting introduces intrinsic irreversibility and a built‑in arrow of time at the fundamental level, rather than treating irreversibility as merely statistical.[1][4]

## Unification postulate

9. Thermodynamics, relativity, and quantum theory form a single entropic continuum  

   - When expressed in the entropic–informational language of ToE, Einstein’s field equations appear as a limiting, geometric case of the more general entropic dynamics.[2][4]

   - Other “gravity from entropy” approaches (e.g., Bianconi‑style models) are recovered as special instances within this broader entropic framework.[4]

In our next exposition, we shall undertake to wade through the sophisticated mathematical foundations of the above the above Postulates.

What is the Theory of Entropicity (ToE)? A Simple Expository Essay on a Radical and Audacious Theory in Modern Theoretical Physics 

The Theory of Entropicity (ToE), first formulated and further developed by John Onimisi Obidi, is a proposed unifying framework in which **entropy** and information are treated as the fundamental physical reality, with spacetime, matter, fields, and forces emerging from entropic and informational dynamics rather than existing as primary entities.[1][2][4][5][6]

## Core idea

In ToE, entropy is promoted from a statistical bookkeeping quantity to an active field $$S(x)$$ (or an entropic field $$\Phi_E(x^\mu)$$) whose gradients and flow drive all physical phenomena.[4][5][6] The visible universe is interpreted as a thermodynamic “projection” or image of an underlying informational manifold, where information and its entropic evolution generate what we experience as geometry, energy, and matter.[1][5][6]

## Entropy as the fundamental field

ToE postulates that all interactions, motion, and apparent curvature arise from the flow and redistribution of entropy, not from intrinsic forces or a fundamentally curved spacetime.[4][5]

- Objects do not intrinsically attract or repel; instead, they follow paths dictated by entropy gradients and maximization.[4]

- Spacetime does not fundamentally curve on its own; the appearance of curvature is produced by variations and curvature in the entropic field.[4][1]

- Forces are not independent entities but emergent descriptions of how systems evolve along entropy-driven optimal paths.[4][5]

An illustrative example is the reinterpretation of gravity: rather than being a fundamental force (Newton) or pure spacetime curvature (Einstein), gravity is treated as an emergent phenomenon arising from entropic constraints and gradients in the entropic field.[2][4]

## Key mathematical and conceptual structures

Several structural elements are highlighted in the Theory of Entropicity program:

- A bridging relation of the form $$\hbar c = k_B T_S \ell_S$$ links quantum $$\hbar$$, relativistic $$c$$, and thermodynamic $$k_B$$ constants via an “entropic temperature” and length scale, providing a bridge between information, energy, and curvature.[1]

- The Obidi Action is introduced as an informational/entropic action principle from which field equations are derived, analogous to how the Einstein–Hilbert action generates Einstein’s equations.[1][5]

- An Informational–Geometric Field Equation generalizes Einstein’s field equation into an informational-entropic context, treating geometry as emergent from informational curvature and entropy flow.[1][5]

- The framework uses information geometry (e.g., Čencov-type connections) to give a rigorous mathematical foundation to entropy-driven dynamics.[5][6]

These structures aim to unify thermodynamics, relativity, and quantum theory into one entropic/informational continuum.[1][5][6]

## Physical implications and examples

The ToE program has been applied to several phenomena to demonstrate its explanatory power:

- Mercury’s perihelion precession is re-derived using entropy-corrected Newtonian gravity (via an Entropic Force-Field Hypothesis), recovering the same 43 arcseconds per century that Einstein obtained, but attributing the effect to entropy constraints rather than spacetime curvature.[2][4]

- Relativistic effects such as mass increase, time dilation, and length contraction are reinterpreted as consequences of finite entropy propagation, entropic conservation, and redistribution—encoded in principles like the Entropic Resistance Principle, the Entropic Resistance Field, the Entropic Cone, and the No-Rush Theorem—rather than being postulated kinematically from spacetime geometry.[5][7]

- Phenomena like the Casimir effect, inertial mass, and gravity are framed as expressions of entropic curvature and informational temperature, without appealing to virtual particles or a pre-given quantum vacuum.[1][5]

A simple way to picture this is: instead of starting from spacetime and putting matter and fields in it, ToE starts from an underlying informational–entropic “substrate,” and shows how what we call spacetime, matter, and forces are efficient macroscopic encodings of how entropy flows and redistributes.[1][4][5][6]

## Philosophical position

Philosophically, the Theory of Entropicity treats information and entropy as ontologically primary, with the laws of physics emerging as constraints on self-organizing entropy flow in an informational continuum.[1][6] This places ToE in the broader family of informational and entropic approaches to fundamental physics, but it distinguishes itself by:

- Elevating entropy to a genuine field and causal substrate rather than a statistical descriptor.  

- Providing explicit entropic action principles and field equations intended to reproduce and generalize known gravitational and relativistic results.[1][2][4][5]

In our next exposition, we shall walk you through one concrete derivation (for example, the entropic derivation of Mercury’s perihelion shift or the entropic reinterpretation of the Lorentz factor) in more mathematical detail.

Citations:

[1] From the Temperature of Information to ... https://www.cambridge.org/engage/coe/article-details/69543375098cdc781fdccf9e

[2] The Theory of Entropicity (ToE): An Entropy-Driven ... https://www.cambridge.org/engage/coe/article-details/67e63abe6dde43c9086de9e0

[3] Entropy - Wikipedia https://en.wikipedia.org/wiki/Entropy

[4] The Theory of Entropicity (ToE): An Entropy-Driven https://www.cambridge.org/engage/api-gateway/coe/assets/orp/resource/item/67e63abe6dde43c9086de9e0/original/the-theory-of-entropicity-to-e-an-entropy-driven-derivation-of-mercury-s-perihelion-precession-beyond-einstein-s-curved-spacetime-in-general-relativity-gr.pdf

[5] The Theory of Entropicity (ToE) Derives and Explains Mass ... https://www.cambridge.org/engage/api-gateway/coe/assets/orp/resource/item/6900d89c113cc7cfff94ef3a/original/the-theory-of-entropicity-to-e-derives-and-explains-mass-increase-time-dilation-and-length-contraction-in-einstein-s-theory-of-relativity-to-r-to-e-applies-logical-entropic-concepts-and-principles-to-verify-einstein-s-relativity.pdf

[6] A Simple Explanation of the Unifying Mathematical ... https://www.authorea.com/users/896400/articles/1348176-a-simple-explanation-of-the-unifying-mathematical-architecture-of-the-theory-of-entropicity-toe-crucial-elements-of-toe-as-a-field-theory

[7] The Theory of Entropicity (ToE) Derives and Explains Mass ... https://papers.ssrn.com/sol3/papers.cfm?abstract_id=5673430

Ontodynamics and Entropology in the Theory of Entropicity (ToE): On Physical Dynamics and Cognitive Dynamics 

Ontodynamics, within the context of the Theory of Entropicity (ToE) developed by John Onimisi Obidi, refers to a theoretical framework where entropy is elevated from a mere statistical measure of disorder to a fundamental, dynamic, "ontic" field that acts as the primary substrate of physical reality. In this view, existence (Being) is defined as the persistence of entropic gradients within finite bounds, while change and motion (Becoming) constitute the continuous, irreversible redistribution of entropy. 

Here are the key aspects of Ontodynamics within the Theory of Entropicity (ToE): 

• Entropy as a Primary Field: Unlike conventional physics, where entropy is a derivative property of matter and energy, ToE posits that the entropy field
  

  $S\left(x\right)$
  is the underlying mechanism from which spacetime, gravitation, and matter emerge.
• Ontic vs. Epistemic: ToE asserts that entropy is an ontic quantity—a real, physical field—rather than merely an epistemic measure of uncertainty.
• The Obidi Action: Ontodynamics is guided by a variational principle known as the Obidi Action, which determines how the entropic field optimizes its flow.
• Emergent Gravity and Motion: Gravity is not considered a fundamental force, but rather an emergent phenomenon arising from the statistical tendency of the entropic field to maximize entropy, causing matter to move towards regions of higher entropy density.
• The No-Rush Theorem: This principle states that physical interactions take a non-zero time for the entropic field to reorganize, setting a finite bound on causality and defining the speed of light as the maximum rate of entropic rearrangement.
• Triadic Information Geometry: The theory connects the physical, informational, and geometric aspects of the universe using a combination of the Fisher-Rao metric (classical), Fubini-Study metric (quantum), and Amari–Čencov
  

  $\alpha$
  -connections, which encode the irreversible flow of entropy.
• Entropology: ToE introduces this term to describe the physics of knowing, where consciousness and perception are considered specialized forms of entropic negotiation. 

The Theory of Entropicity (ToE) is a recent (late 2025) theoretical proposal aiming to unify thermodynamics, quantum mechanics, and relativity by centering all physical laws on the dynamic, ontic field of entropy.

Obidi's Ontodynamics as the Philosophical Foundation of the Theory of Entropicity (ToE)

Obidi's Ontodynamics is a philosophical framework developed by John Onimisi Obidi in the Theory of Entropicity (ToE). It posits that entropy is not just a measure of disorder but a fundamental field that governs all interactions in the universe. 

Key concepts include:

• Ontodynamics: The study of how existence/phenomena/interactions/measurements/observations evolve through entropic dynamics.
• Entropy as a Field: Entropy is treated as a continuous scalar field, replacing traditional notions of mass, energy, and spacetime.
• Emergence of Physical Laws: Physical laws are emergent properties arising from the dynamics of the entropic field.
• Mathematical Framework: The theory is mathematically derived from the Obidi Action, which governs the evolution of the entropic field.
  This framework unifies thermodynamics, general relativity, and quantum mechanics, suggesting that all phenomena, including motion and consciousness, are emergent from the dynamics of entropy. 

References

Home · Entropicity/Theory-of-Entropicity-ToE Wiki

GitHub Wiki on the Theory of Entropicity (ToE): https://github.com/Entropicity/Theory-of-Entropicity-ToE/wiki