On the prismatization of OK beyond the Hodge-Tate locus

Let X=Spf(OK). We classify perfect complexes of n-truncated prismatic crystals on the prismatic site of X when n≤ 1+p-1/e by studying perfect complexes on the n-truncated prismatization of X, which are certain nilpotent thickenings of the Hodge-Tate stack of X inside the prismatization of X. We describe the category of continuous semilinear representations and their cohomology for GK with coefficients in BdR,n⁺ via rationalization of vector bundles on the slight shrinking of the n-truncated prismatization of X. Along the way, we classify certain integral models for de Rham prismatic crystals studied in arXiv:2205.14914.