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