This paper establishes the ultimate mathematical framework of geometric information dynamics, completely categorifying the hierarchical structure of informationinto higher ∞-toposes. The core innovation lies in proving that when informationdynamics is equipped with a complete (∞,1)-categorical structure, the emergenceof spacetime, the dynamics of gravity, and the unitarity of quantum mechanicsall necessarily appear as different facets of the same universal construction. Weconstruct a “spectral ∞-topos”– a mathematical object unifying spectral chaindynamics, quantum field theory, and higher logic– and prove that its classifyingspace of global sections naturally parametrises all physical theories satisfying theinformation axioms, while the Einstein equations correspond to the tangent vectorfield equations of this classifying space. This framework realizes a rigorous formof the Baez–Dolan “physicist’s topologicalization conjecture”, provides the mathematically most complete expression of “it from bit”, and predicts higher topologicalorder phases beyond traditional quantum field theory and their applications inquantum computation.