A partially non-proper ordinal beyond L(V λ+1)

In his recent work, Woodin has defined new axioms stronger than I0 (the existence of an elementary embedding from to itself), that involve elementary embeddings between slightly larger models. There is a natural correspondence between I0 and Determinacy, but to extend this correspondence in the new framework we must insist that these elementary embeddings are proper. Previous results validated the definition, showing that there exist elementary embeddings that are not proper, but it was still open whether properness was determined by the structure of the underlying model or not. This paper proves that this is not the case, defining a model that generates both proper and non-proper elementary embeddings, and compare this new model to the older ones.