On the Geometry and Topology of Information Spaces

We introduce a new mathematical structure, the information space, defined by four axioms that encode the fundamental properties of quantum information: (0) the existence of a qubit with state space CP¹; (1) the determination of a spatial distance between qubits by their quantum mutual information; (2) the emergence of a local time scale from the operational step count of entanglement renewal; and (3) a maximum entropy variational principle governing the dynamics of the network. These axioms are purely information-theoretic and involve no geometric or physical constants. We construct the category of information spaces, whose objects are locally ringed spaces equipped with an entanglement distance, and whose morphisms are distance-nonincreasing morphisms of locally ringed spaces. The fundamental result is the existence of a contravariant spectral functor: to the category of perfectoid spaces, which is fully faithful and essentially surjective, establishing a categorical equivalence. This equivalence reveals that the geometry and topology of information spaces are encoded in the perfectoid spaces of Scholze. Conversely, every perfectoid space admits an information-theoretic description in terms of entanglement distances. As applications, we develop the homotopy theory, cohomology, and classification of information spaces, and we demonstrate how the Langlands program emerges naturally from the algebraic-geometric binarity of these spaces. The present paper is the first in a series devoted to the systematic study of information spaces as autonomous mathematical objects.