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Abstract
This ToE Letter IE establishes that the Riemannian curvature of physical spacetime is not a primitive geometric datum posited a priori, but rather emerges as the macroscopic, thermodynamic-limit expression of curvature defined on an underlying statistical-information manifold. Working within the axiomatic framework of the Theory of Entropicity (ToE), we construct the information manifold (ℳ_I, gI) from the Fisher–Entropic metric on a fundamental entropic substrate Ω, define its intrinsic Riemann curvature tensor, and prove a Curvature Transfer Theorem demonstrating that the spacetime Riemann tensor RS is the pushforward of the information Riemann tensor RI in the thermodynamic limit. Einstein's field equations [1] are thereby recovered as an emergent identity rather than a fundamental law. We introduce the Obidi Curvature Invariant (OCI) 𝒦_Ω — a non-negative scalar field measuring the residual information curvature not captured by spacetime geometry — and establish its key properties: vanishing in the classical limit, positivity, gauge invariance, and a topological bound. The invariant 𝒦_Ω identifies the informational degrees of freedom relevant to quantum gravity and may contribute to the effective cosmological constant. The purpose of this comprehensive Preamble is to provide the reader with self-contained explanation of why the three principal structures of information geometry employed in the formulation of the Theory of Entropicity (ToE) — the Fisher–Rao metric, the Fubini–Study metric, Amari–Čencov α-connections — are not merely convenient mathematical tools borrowed from statistics and quantum information theory, but are instead the authentic geometric substrates from which the physical universe emerges in the Theory of Entropicity (ToE).
