Non-tame mice from tame failures of the unique branch hypothesis (Q2876534)

\textit{D. A. Martin} and \textit{J. R. Steel}, in their proof of projective determinacy from Woodin cardinals [J. Am. Math. Soc. 2, No.1, 71-125 (1989; Zbl 0668.03021)], introduced the Unique Branch Hypothesis (UBH). This hypothesis states that every iteration tree that acts on the universe of sets \(V\) has at most one cofinal branch. Assuming that there is a proper class of strong cardinals, the authors show that the failure of the unique branch hypothesis for tame trees implies that there is a set generic extension of \(V\) in which there is a transitive inner model \(M\), containing \(\mathbb{R}\), such that \(M\models \text{AD}^++\Theta>\theta_0\). Here, \(\Theta\) is the supremum of the ordinals that are the surjective image of \(\mathbb{R}\), and \(\theta_0\) is the the supremum of the ordinals that are the surjective image of \(\mathbb{R}\) by an ordinal-definable function. The authors use a core model induction technique, a method discovered by W. H. Woodin that is used to construct models of determinacy.