Non-tame mice from tame failures of the unique branch hypothesis

TL;DR

This paper demonstrates that failures of the unique branch hypothesis for tame trees imply the existence of non-tame mice, extending previous results and connecting large cardinal hypotheses with inner model theory.

Contribution

It establishes a link between UBH failures for tame trees and the existence of non-tame mice, extending Steel's earlier work.

Findings

Failure of UBH for tame trees implies non-tame mice existence.

In some generic extension, a model satisfies AD+ with Θ > θ₀.

Results extend Steel's earlier findings on tame trees.

Abstract

In this paper, we show that the failure of the unique branch hypothesis (UBH) for tame trees (see \rdef{tame iteration tree}) implies that in some homogenous generic extension of $V$ there is a transitive model $M$ containing $Ord \cup \mathbb{R}$ such that $M\vDash AD^+ + \Theta > \theta_0$. In particular, this implies the existence (in $V$) of a non-tame mouse. The results of this paper significantly extend Steel's earlier results from \cite{steel2002core} for tame trees.