Tilting equivalence of finite almost derived algebraic cobordism for perfectoid algebras

We prove tilting equivalence for the finite almost derived algebraic cobordism spectrum ${\mathrm{d}\mathrm{M}\mathrm{G}\mathrm{L}}^{a,fin}$ of perfectoid algebras. More precisely, for any perfectoid algebra, the tilting functor induces a weak equivalence on the corresponding finite almost derived algebraic cobordism spectra. This invariant is a finite-syntomic, derived, and non-$\text{msthbb}{A}^1$-local version of algebraic cobordism, designed to retain infinitesimal deformation data over mixed-characteristic bases. To prove the result, we isolate a form of excisive approximation for pointed infinity-categories, including non-presentable ones. In the locally finitely presentable case, this agrees with Heuts’s framework. We also define approximation functors along natural transformations and apply them to algebraic cobordism and $K$-theory, obtaining results such as Bott periodicity and Gabber rigidity.