Descent of $(\phi ,\tau )$-modules in characteristic $p$

In this article, we study the descent of $(\phi ,\tau )$-modules over perfectoid period rings in characteristic $p$ via Berger and Rozensztajn’s theory of super-Hölder vectors. This is a generalization of their work on $(\phi ,\mathrm{\Gamma })$-modules. As an application, we answer a question of Caruso regarding the connection between $(\phi ,\tau )$-modules and $(\phi ,\mathrm{\Gamma })$-modules without involving Galois representations as intermediaries.