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