This paper develops a complete and rigorous mathematical foundation for the information Ricci flow on the infinite-dimensional manifold of smooth positive probability densities on a compact Riemannian manifold. We construct the Fisher--Rao metric, derive the Levi--Civita connection and geodesic equation by explicit variational calculus, and establish local well-posedness in Sobolev spaces. We then introduce an information entropy functional of Perelman type, compute its first variations with respect to both the metric and the density, and derive the information Ricci flow as a gradient flow. A detailed monotonicity formula is obtained. We define information singularities, information horizons, and an information area law. All derivations are original and presented in full detail, providing a rigorous mathematical basis for the principle ``information is matter''.
yuanjian li (Mon,) studied this question.