A stationary-tower-free proof of the derived model theorem

Woodin proved the theorem in perhaps 1986 or 1987, using stationary tower forcing and (through the work of Martin and the author) iteration trees. The proof we give here uses only iteration trees. Stationary tower forcing is replaced by “genericity iterations”, as it can be in certain related contexts as well. We believe that the unity of method in the resulting proof gives it some interest. The first stationary-tower-free proof of the special case L(R∗) |= AD was discovered by the author in the early 90’s. That proof uses fine-structural mice and Woodin-style genericity iterations to replace the stationary tower forcing. The fine-structural mice were needed because Woodin-style genericity iterations require ω1 +1-iterable structures, and we do not know how to prove that kind of iterability for countable M ≺ V . In the mid 90’s, Neeman found a new kind of genericity iteration that requires only ω + 1 iterabilty, which