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 I₈₉ and derive an information--Einstein equation₈₉=\, I₈₉[. \]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.