  Let $p:X \rightarrow S$ be a flat, proper and regular scheme over a strictly
henselian discrete valuation ring. We prove that the singularity category of
the special fiber with its natural two-periodic structure allows to recover the
$\ell$-adic vanishing cohomology of $p$. Along the way, we compute
homotopy-invariant non-connective algebraic K-theory with compact support of
certain embeddings $X_t \hookrightarrow X_T$ in terms of the motivic
realization of the dg category of relatively perfect complexes.
