Information as Matter: A Unified Information-Geometric Framework for Quantum Theory, Thermodynamics, and Gravity

This work develops a unified information-geometric framework in which quantum theory, thermodynamics, and gravityemerge as different manifestations of a single underlying structure: the geometry and dynamics of an informationmanifold. The central postulate is that physical reality can be encoded in a smooth manifold of distinguishable states equipped with an information metric and compatible connections. On this manifold we construct a unified action functional that couples information curvature, quantum Fisher information, and information flows. From this action we derive, by explicit and rigorous variational calculations, a family of original field equations that we identify and label as the core equations of the Information-as-Matter Unified Framework (IMUF). These include: (i) an information-driven Einstein equation, where the source term is an information stress tensor built from gradients of the log-density; (ii) a quantum--thermal unified evolution equation for density operators, combining unitary quantum dynamics with information-geometric gradient flows; (iii) an information Ricci flow equation in which the metric evolves under both curvature and information gradients; (iv) a curvature-based entanglement lawrelating entanglement entropy to information curvature; (v) a unified fluctuation relation in which statistical and geometric contributions to fluctuations are combined; and (vi) an information-theoretic generalization of the black hole area law. We discuss the mathematical structure of these equations, their relation to known theories, and a set of concrete physical predictions and possible observational or numerical tests. The resulting framework realizes, in a precise and calculable sense, the slogan ``information is matter''.