Floor, ceiling, slopes, and K-theory

We calculate K_*(k[x]/xᵉ; Zₚ) by evaluating the syntomic cohomology Zₚ(i)(k[x]/xᵉ) introduced by Bhatt-Morrow-Scholze and Bhatt-Scholze. This recovers calculations of Hesselholt-Madsen and Speirs, and generalizes an example of Mathew treating the case $e=2$ and $p>2$. Our main innovation is systematic use of the floor and ceiling functions, which clarifies matters substantially even for $e=2$. We furthermore observe a persistent phenomenon of slopes. As an application, we answer some questions of Hesselholt.