Even periodization of spectral stacks

We introduce the operation of even periodization on nonconnective spectral stacks. We show how to recover from it the even filtration of Hahn-Raksit-Wilson, and (a Nygaard-completion of) the filtered prismatization stack of Bhatt-Lurie and Drinfeld. We prove that the moduli stack of oriented elliptic curves of Goerss-Hopkins-Miller and Lurie is the even periodization of the spectrum of topological modular forms. Likewise, the chromatic moduli stack, which we previously studied under the name of the moduli stack of oriented formal groups, arises via even periodization from the sphere spectrum. Over the complex bordism spectrum, we relate even periodization with the symmetric monoidal shearing, in the sense of Devalapurkar, of free gradings.