Motivic Steenrod operations at the characteristic via infinite ramification

We construct motivic power operations on the mod-$p$ motivic cohomology of ${\text{Fb}}_p$-schemes using a motivic refinement of Nizioł’s theorem. The key input is a purity theorem for motivic cohomology established by Levine. Our operations satisfy the expected properties (naturality, Adem relations, and the Cartan formula) for all bidegrees, generalizing previous results of Primozic which were only know along the ``Chow diagonal.‘’ We offer geometric applications of our construction: 1) an example of non-(quasi-)smoothable algebraic cycle at the characteristic, 2) an answer to the motivic Steenrod problem at the characteristic, 3) a counterexample to the integral version of a crystalline Tate conjecture.