Iterates of $M_1$

Assume ${\mathbf{\Delta }}_2^1$-determinacy. Let $L_{{\kappa }_3}[T_2]$ be the admissible closure of the Martin-Solovay tree and let $M_{1,\mathrm{\infty }}$ be the direct limit of $M_1$ via countable trees. We show that $L_{\kappa_3}[T_2] \cap V_{u_{\omega}} = M_{1,\infty} | u_{\omega}$.