We develop a detailed and rigorous theory of the information Ricci flow on the infinite-dimensionalmanifold of smooth positive probability densities on a compact Riemannian manifold. Starting fromfirst principles, we construct the information manifold as a Fr\'echet and Sobolev Hilbert manifold,define the Fisher--Rao metric, and derive the Levi--Civita connection and geodesic equation by explicitvariational calculus and by the Koszul formula. We then introduce an information entropy functionalof Perelman type, compute its first variations with respect to both the metric and the density in fulldetail, and derive the information Ricci flow as a gradient flow in a suitable formal sense. We providea careful analysis of the monotonicity of the entropy functional along the flow, including the structureof the quadratic terms and the role of the information coupling. Finally, we define informationsingularities, information horizons, and an information area law, and discuss their conceptualinterpretation in the ``information is matter'' paradigm. The paper is self-contained and writtenwith full derivations, aiming at a mathematically complete foundation for infinite-dimensionalinformation geometry and its Ricci-type flows.
Y. Li (Mon,) studied this question.