n^{\pm j } ) (\delta j) \Bigr| \\
    \begin{split}
    &\lesssim 
    \| {\operatorname{I}\!}v^j \|_{\dot{H}^1} \| {\operatorname{I}\!}( u^j + v^j) \|_{\dot{H}^1} + \| {\widetilde}{m}^{\pm j}\|_{L^2} \|  n^{\pm j} + {\widetilde}{m}^{\pm j}\|_{L^2} \\
    &\quad + \| u^{j} + v^{j} \|_{L^2} \| u^{j} + v^{j}\|_{H^{\frac12+}} \| {\widetilde}{m}^{\pm j} \|_{L^2} + \Bigl( \| u^{j} + v^{j}\|_{L^2} \| v \|_{H^{\frac12+}} + \| u\|_{L^2} \|v\|_{H^{\frac12+}} \Bigr) \| n^{\pm j} \|_{L^2}
    \end{split}\\
    &\lesssim 
    {N_{\text{I}}}^{1-s} {N_{\text{HL}}}^{s-\frac12 -\beta  } {N}. \end{aligned}$$

Thus the total growth of the Hamiltonian over $[0, \delta J]$ is bounded by $$\begin{aligned}
{N_{\text{I}}}^{-1+} {N}^3 
+ (J-1) \Bigl( 
    {N_{\text{I}}}^{-1+} {N}^2  + 
    {N_{\text{I}}}^{-2+}{N}^4 + {N_{\text{I}}}^{-\frac32+}{N}^{3} + 
    {N_{\text{I}}}^{1-s} {N_{\text{HL}}}^{s-\frac12 -\beta   }{N}\Bigr)  \\
    = {N}^2 \Bigl[
    {N_{\text{I}}}^{-1+} {N}+ 
    (J-1)\Bigl( 
    {N_{\text{I}}}^{-1+}   + 
    {N_{\text{I}}}^{-2+}{N}^2 + {N_{\text{I}}}^{-\frac32+}{N}+ {N_{\text{I}}}^{1-s} {N_{\text{HL}}}^{s-\frac12 -\beta   } {N}^{-1} \Bigr) \Bigl].  \end{aligned}$$ This is acceptable as long as ${N_{\text{I}}}^{-1+} {N}\lesssim 1$ and $$J \lesssim \min\left\{ {N_{\text{I}}}^{1-}  ,\;\; {N_{\text{I}}}^{2-} {N}^{-2},\;\; {N_{\text{I}}}^{\frac32 -} {N}^{-1},\;\; {N_{\text{I}}}^{-(1-s)} {N_{\text{HL}}}^{-(s-\frac12 -\beta   )} {N}\right\} .$$ For such $J$, we can iterate to cover an interval of length $$\delta J \approx  \min\left\{ {N_{\text{I}}}^{1-} {N}^{-2-}  ,\;\;  {N_{\text{I}}}^{2-} {N}^{-4-}, \;\; {N_{\text{I}}}^{\frac32 -} {N}^{-3-}, \;\; {N_{\text{I}}}^{-(1-s)} {N_{\text{HL}}}^{-(s-\frac12 -\beta   )} {N}^{-1-} \right\}.$$ Recalling that $$N \approx  \max\{ K{N_{\text{I}}}^{1-s} , C_1 {N_{\text{HL}}}^{\frac12 -\beta   }\},$$ we obtain the following bounds. To simplify the expressions, 