Thin equivalence relations in scaled pointclasses

For ordinals α beginning a Σ 1 gap in \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathrm{L}(\mathbb {R})$\end{document} , where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma _{1}^{\mathrm{J}_{\alpha }(\mathbb {R})}$\end{document} is closed under number quantification, we give an inner model‐theoretic proof that every thin \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Sigma _{1}^{\mathrm{J}_{\alpha }(\mathbb {R})}$\end{document} equivalence relation is \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\Delta _{1}^{\mathrm{J}_{\alpha }(\mathbb {R})}$\end{document} in a real parameter from the (optimal) hypothesis \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathsf {AD}^{\mathrm{J}_{\alpha }(\mathbb {R})}$\end{document} .