Unstable arithmetic fracture squares in $\infty$-topoi

We show that for a large class of $\infty$-topoi there exist unstable arithmetic fracture squares, i.e. squares which recover a nilpotent sheaf $F$ as the pullback of the rationalization of $F$ with the product of the $p$-completions of $F$ ranging over all primes $p\in\mathbb Z$.