From Information Geometry to Information Gravity Information Geometry as the Origin of Einstein’s Gravity: Correspondence of the Obidi Action and the Einstein–Hilbert Action in the Theory of Entropicity (ToE), Theory of Entropicity (ToE) Living Review Letters Series (ToE LRLS) — Letter III

Obidi shows that Lorentzian spacetime geometry emerges from information geometry via a controlled entropy‑gradient disformal transformation, and that the curvature of this emergent metric reproduces the Einstein gravity of General Relativity (GR). This monograph establishes, with full mathematical and conceptual rigor, the deep structural correspondence between the Obidi Action of the Theory of Entropicity (ToE) and the Einstein–Hilbert Action of General Relativity (GR). We demonstrate that information geometry — comprising the Fisher–Rao metric of classical statistical distinguishability, the Fubini–Study metric of quantum state distinguishability, and the Amari–Čencov α-connections — constitutes the hidden geometric substratum from which gravitational dynamics emerge. The Obidi Action, the variational centerpiece of ToE, accomplishes for the entropic field precisely what the Einstein–Hilbert Action accomplishes for spacetime curvature: it converts a geometric manifold into a dynamical physical arena. We trace the complete logical and mathematical pipeline — from the ontological entropy field S(Λ) through the Hybrid Metric-Affine Space (HMAS), through the α–q constitutive constraint linking Rényi–Tsallis functional deformation to affine geometric asymmetry, and through the emergence of the Master Entropic Equation (MEE) — ultimately recovering Einstein's field equations as the low-gradient, near-equilibrium limit of the entropic field equations. The Einstein–Hilbert Action is thus subsumed within the Obidi Action as a special case, establishing gravity not as a fundamental geometric postulate but as an emergent consequence of information-geometric dynamics. We further examine the Vuli–Ndlela Integral, irreversibility and the arrow of time into the framework. Reviewed landscape of entropic gravity — Jacobson's thermodynamic, Verlinde's gravity, and Bianconi's Gravity.