How does the Theory of Entropicity (ToE) make the Amari-Čencov α-connections and Fisher-Rao and Fubini-Study Metrics to become physical spacetime from which Einstein field equations emerge?

In the Theory of Entropicity (ToE), the **Amari–Čencov $$\alpha$$**-connections, the **Fisher–Rao metric**, and the **Fubini‑Study metric** do not stay abstract tools of information geometry; they are **promoted to physical spacetime structures** through an entropy‑driven transformation, and from that transformation Einstein’s field equations are posited to emerge as a limiting case [7][9][2].

### How information geometry becomes physical spacetime

1. **Entropic manifold as the physical arena**  

   ToE treats the universe as an **entropic manifold**: a space of informational states where the “distance” between nearby states is measured by Fisher–Rao (classical information) and Fubini–Study (quantum information) [7][6]. Ordinarily these metrics only quantify distinguishability of distributions or quantum states; ToE says they actually encode **physical distances and intervals** in the deeper entropic substrate [5][2].

2. **Fisher–Rao and Fubini‑Study become the spacetime metric**  

   The Fisher–Rao metric is identified with the **classical‑limit piece of the spacetime metric**, while Fubini‑Study corresponds to the **quantum‑fluctuation layer** of the same geometry [4][5][6]. A “metric‑transformation” scheme is introduced so that the physical metric $$g_{\mu\nu}$$ is a deformation of the Fisher–Rao / Fubini‑Study information metric by a factor depending on the entropic field $$S(x,t)$$ and the $$\alpha$$‑index [5][9][2]. Symbolically, the literature sketches a relation like  

   $$

      g_{\mu\nu}^{\text{(phys)}} \sim \Phi(\alpha, S)\, g_{\mu\nu}^{\text{(FR/FS)}}

   $$

   where the scalar field $$\Phi$$ ties entropy and information together.

3. **$$\alpha$$-connections become the affine connection**  

   The Amari–Čencov $$\alpha$$-connection is no longer a formal object in statistical models; it is taken as the **physical affine connection** of the entropic spacetime, entering the Obidi Field Equations (OFE) directly as the geometric part that tells how vectors and geodesics evolve under entropic gradients [7][5]. The $$\alpha$$‑index then becomes a physical deformation parameter, tied to non‑extensive entropy via $$\alpha = 2(1 - q)$$, so that **affine asymmetry** reflects **irreversible information‑entropy flow** [7][9].

### How Einstein’s equations emerge

4. **Obidi Action and entropic curvature**  

   The Obidi Action is a variational principle for the entropic field $$S(x,t)$$, which yields the **Master Entropic Equation (MEE)**, entropic geodesics, and an entropy potential equation [7][9]. In this framework, the Fisher–Rao–type curvature and the $$\alpha$$-connection curvature are interpreted as **physical spacetime curvature**: the same curvature that in general relativity is encoded in the Ricci tensor and Einstein tensor [7][2].

5. **From information curvature to Einstein curvature**  

   ToE argues that, in an appropriate limit, the MEE maps onto equations structurally equivalent to the Einstein field equations, with “entropic‑stress” objects playing the role of the energy‑momentum tensor $$T_{\mu\nu}$$ [7][9][11]. Photon‑deflection and perihelion‑shift results are then reproduced using entropy‑constrained geodesics, without starting from a metric postulate, and the numerical agreement with GR is cited as evidence that information‑geometric curvature has become **effective gravitational curvature** [1][2].

6. **Geometrization of entropy à la Einstein**  

   In this picture, spacetime is not assumed; it is **reconstructed** from the Fisher–Rao / Fubini‑Study metrics and the $$\alpha$$-connection, all deformed by the entropic field, and Einstein’s equations are then presented as the **macroscopic geometric expression** of that underlying entropic information structure [7][9][6]. ToE thus frames itself as continuing Einstein’s “geometrization of physics,” but with **entropy replacing matter** as the primary ontological field whose curvature generates geometry and gravity [7][2].