Characterize structures M0 for which the reduced power M0^N/Fin has stable theory

Identify and characterize all first-order structures M_0 such that the reduced power M_0^N/Fin has a stable first-order theory, in terms of properties of Th(M_0).

Background

The Feferman–Vaught framework implies that the stability of the reduced product’s theory depends only on the theories of the factors, but translating this into a usable structural criterion on a single base structure M_0 is not straightforward.

The authors emphasize the lack of a convenient characterization for when the reduced power of a single structure over Fin is stable, highlighting subtleties illustrated by examples where stability behavior changes under reduced powers.