Crystalline Cohomology of Diamond Spectra via Perfectoid Shimura Varieties

We construct a robust theory of crystalline cohomology for diamond spectra arising from the transfinite inverse limit of Shimura varieties with infinite level structure at p. By invoking the theory of prismatization developed by Bhatt and Scholze, adapted to the context of non-archimedean analytic geometry over transfinite ordinals, we establish a comparison isomorphism between the pro-étale cohomology of the diamond S⋄ and the crystalline cohomology of its special fiber, twisted by the period ring Ainf . We prove that for a perfectoid Shimura variety of Hodge type, the associated diamond spectrum admits a canonical filtration compatible with the Hodge-Tate decomposition. This re- solves the "Weight-Monodromy" conjecture in the limit of infinite level structures for the super-rigid analytic locus.