A Proof of the Langlands Correspondence via Information Geometry

We establish a categorical equivalence between the information space category $\Info$---whose objects are locally ringed spaces equipped with an entanglement distance structure derived from quantum mutual information---and the perfectoid space category $\Perfd$ in the sense of Scholze. The equivalence is realized by a contravariant spectral functor $\D: \Info \to \Perfd$ that assigns to each information space its spectral space. We prove that $\D$ is fully faithful and essentially surjective (Theorem~A). Within this framework, we prove the geometric Langlands duality for all reductive groups over $p$-adic fields as the fibration of algebraic versus geometric coordinates on the $G$-information space $X_p^G$ (Theorem~B). Extending this duality to the global setting via the restricted product of local valued information spaces, we establish the global Langlands correspondence for all reductive groups over all number fields---the central conjecture of the Langlands program---as an equivalence between the category of global Galois representations and the category of global automorphic forms (Theorem~C). The proof of essential surjectivity reduces to a purely categorical lemma: the restricted product of local equivalences is an equivalence. We further prove the Langlands functoriality conjecture via quantum state extension, characterize arithmetic $L$-functions as regularized characteristic functions of the entanglement spectrum of associated quantum states, and establish the analytic continuation and functional equation of global $L$-functions within the information space framework. This provides a unified information-theoretic foundation for the entire Langlands program: the dualities between Galois representations and automorphic forms, between $L$-functions and entanglement spectra, and between functorial lifting and quantum state extension all emerge as manifestations of a single underlying principle---the algebraic-geometric binarity of quantum information structures.