The Lange-Stuhler Theorem via Gerbes

We recast the Lange-Stuhler theorem as the universal property of a $2$-cartesian square of classifying stacks. This formulation shows that every Frobenius-periodic vector bundle of rank $r$ on an arbitrary fibered category in characteristic $p>0$ is trivialized by a finite étale torsor, without geometric hypotheses or restrictions on the ground field beyond its characteristic. The construction extends from ${\mathrm{G}\mathrm{L}}_n$ to group schemes over finite fields that are connected affine or of finite type.