Geometry from Information: Uniqueness of Relative Entropy as the Gravitational Potential.

This monograph demonstrates that Einstein's field equations emerge uniquely from the requirement that the information-theoretic distance from the de Sitter vacuum, measured by relative entropy (S_{\mathrm{rel}}), serves as a potential on the space of metrics. The uniqueness follows from four fundamental axioms: monotonicity under quantum channels, additivity on tensor products, normalization, and compatibility with the first law of entanglement thermodynamics. By employing the Bogoliubov-Kubo-Mori (BKM) metric and Helmholtz integrability, we show that the Einstein-Lovelock structure is the only consistent local dynamics for such an information-theoretic potential. The cosmological constant \Lambda is identified as the curvature radius of the information manifold, while Newton's constant G_N measures the information cost of metric deformations.