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Ladder mice
Assume ZF + AD + $V=L(\mathbb{R})$. We prove some "mouse set" theorems, for definability over $J_\alpha(\mathbb{R})$ where $[\alpha,\alpha]$ is a projective-like gap (of $L(\mathbb{R})$) and $\alpha$ is either a successor ordinal or has countable cofinality, but $\alpha\neq\beta+1$ where $\beta$ ends a strong gap. For such ordinals $\alpha$ and integers $n\geq …