norm of its nonlinear part is at most order ${N_{\text{HL}}}^{s-\beta - \frac12  }$. That is, the nonlinear part is small and smooth. [^2]

We add this nonlinear part $(v^1, {\widetilde}{m}^{\pm1})$ to $(u^1(\delta), n^{\pm1}(\delta))$ and evolve according to again to obtain functions $(u^2,n^{\pm2})$ which solve on the interval $[\delta, 2 \delta]$. At time $2\delta$, we again add in the smooth part $(v^2, {\widetilde}{m}^{\pm2})$ of the solution to the difference equation, and again evolve according to obtain $(u^3, n^{\pm3})$. This can continue as long as the norm of the solution does not grow too much. To control the growth, we carefully choose the frequency thresholds ${N_{\text{HL}}}$ and ${N_{\text{I}}}$.

To understand the growth of the norm, we study the Hamiltonian energy. The growth of the Hamiltonian is bounded as follows. Recall that $(u^j,n^{\pm j})$ is defined on the interval $[(j-1)\delta, j \delta]$ with $$(u^{j}, n^{\pm j })(\delta(j-1)) = \Bigl(u^{j-1} + v^{j-1}, n^{\pm j-1 } + {\widetilde}{m}^{\pm j-1 }\Bigr)(\delta(j-1)) .$$ Then the growth of the Hamiltonian is bounded by $$\begin{aligned}
     \Bigl|H({\operatorname{I}\!}u^J, n^{\pm J})(\delta J) - H({\operatorname{I}\!}u^1, n^{\pm 1} )(0)\Bigr| 
    &\leq 
    \Bigl| H({\operatorname{I}\!}u^J, n^{\pm J})(\delta J) - {\widetilde}{H}( u^J, n^{\pm J})(\delta J) \Bigr| \\
    & + \sum_{j=1}^J \Bigl| {\widetilde}{H}( u^j, n^{\pm j})(\delta j) - {\widetilde}{H}( 