We emphasize that we are fixing all the other gluing parameters so that only $\beta$ varies. We will prove some effective estimates in Section 8 and Section 9 which give uniform etimates for the linearized gluing operator $\mathscr{L}$ (defined in Section 1.3) and for sufficiently large $\beta\gg 1$. We also note that the estimates are unifrom as long as other parameters vary in compact sets.