Toward a Unified ``Information = Matter'' Theory Information Manifolds, Stress--Energy, Quantum Extensions, and a Systematic Derivation of Key Equations

We develop a detailed and mathematically explicit framework for a prospective ``information = matter'' unified theory. Starting from the infinite-dimensional information manifold of smooth positive probability densities endowed with the Fisher--Rao metric, we construct an information stress--energy tensor Iᵢⱼ[ρ] and derive an information--Einstein equation\[Gᵢⱼ=κ\,Iᵢⱼ[ρ].\]We then systematically extend this structure in three directions: (i) an effective reconstruction of classical matter fields (scalar fields, fluids, and gauge fields) from information data; (ii) a quantum information manifold based on density operators and quantum Fisher information, leading to a quantum information stress--energy tensor and a quantum information--Einsteinequation; (iii) a unified master equation that couples geometric curvature, classical information, and quantum information. Throughout the paper we explicitly mark and discuss several original equations (denoted {n}) that are proposed as structural candidates for a genuine ``information = matter'' unified theory. We emphasize both what is rigorously derived and what is conjectural or conditional, thereby clarifying the current status and the remaining gaps toward a complete unification.