  We construct well-behaved extensions of the motivic spectra representing
generalized motivic cohomology and connective Balmer--Witt K-theory (among
others) to mixed characteristic Dedekind schemes on which 2 is invertible. As a
consequence we lift the fundamental fiber sequence of $\eta$-periodic motivic
stable homotopy theory established in [arxiv:2005.06778] from fields to
arbitrary base schemes, and use this to determine (among other things) the
$\eta$-periodized algebraic symplectic and SL-cobordism groups of mixed
characteristic Dedekind schemes containing 1/2.
