Building on the work of Schimmerling [Coherent sequences and threads, Adv. Math.216 89–117] and Steel [PFA implies AD L, J. Symbolic Logic70 1255–1296], we show that the failure of square principle at a singular strong limit cardinal implies that there is a nontame mouse. The proof presented is the first inductive step beyond L of the core model induction that is aimed at getting a model of ADℝ + "Θ is regular" from the failure of square at a singular strong (...) limit cardinal or PFA. (shrink)

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The goal of this paper is to compute the full HOD of models of AD+AD+ of the form L)L) below “ADR+ΘADR+Θ is regular”. As part of this computation, we give a computation of HOD|ΘHOD|Θ left open in [3] for Θ a successor in the Solovay sequence. Our work, when combined with [3], shows that in AD+AD+ models of the form L)L) below “ADR+ΘADR+Θ is regular”, HOD satisfies GCHGCH.