Information as Matter in Infinite Dimensions Fisher Geometry, Information Ricci Flow, and a Unified Field Framework

This paper develops a rigorous infinite-dimensional information-geometric framework aimed at realizing the principle ``information is matter'' as a mathematically coherent unification scheme. We construct an infinite-dimensional information manifold of smooth probability densities on a compact Riemannian manifold, equipped with a Fisher--Rao type metric and its Levi--Civita connection. Within this setting we derive geodesic equations, prove local well-posedness of geodesic flows in suitable Sobolev completions, and introduce an information Ricci flow that couples the base manifold geometry with the information density.We define and analyze an information entropy functional of Perelman type, prove its monotonicity along the coupled flow, and identify natural notions of information singularities and information horizons. On this basis we formulate and derive several original equations that implement the ``information as matter'' paradigm in an infinite-dimensional context, including: an information-driven Einstein-type equation, an information Ricci flow equation, a coupled metric--density evolution system, and a family of unified fluctuation and entanglement--curvature relations. These equations provide a purely geometric and information-theoretic description of structures usually attributed to matter and fields, and thus constitute a concrete step toward a unified theory in which matter, geometry, and information are different aspects of a single underlying infinite-dimensional structure.