Six-functor formalism for Kummer étale cohomology of log schemes

We establish a Grothendieck six-functor formalism for Kummer étale cohomology including Poincaré duality for every separated vertical exact log smooth morphism of noetherian fs log schemes $f\colon X\rightarrow S$ when the coefficient ring $Λ$ is killed by an integer invertible on $S$. This is done via log étale rigidity \[\mathrm{D}_{\mathrm{l\acute{e}t}}(S,Λ)\simeq \mathrm{DA}_{\mathrm{l\acute{e}t}}(S,Λ).\] To achieve this, we also prove that Kummer étale cohomology satisfies $\mathbb{A}^1$-invariance, invariance under virtual isomorphisms, log cdh-descent, and invariance under verticalization.