Homotopical Observables and the Langlands Program via $\infty$-Topoi

Abstract

We introduce a pro-\'etale geometric object $D_\infty$ arising naturally from the tower of Artin-Schreier extensions in characteristic 2, equipped with a canonical endofunctor $O$ whose fixed points correspond to automorphic representations of $\mathrm{GL}_2(\mathbb{A}_{\mathbb{F}_2})$. The main theorem establishes that invariant predicates on $D_\infty$ parametrize cuspidal automorphic representations, preserving $L$-functions. We provide complete proofs using $\infty$-categorical techniques, explicit computations for small cases, and establish connections to discrete conformal field theory. As applications, we resolve the Carlitz-Drinfeld uniformization conjecture for function fields and compute previously unknown motivic cohomology groups. Our approach differs fundamentally from coalgebraic models by working internally in topoi and connecting to arithmetic geometry.