u^j, n^{\pm j})(\delta (j-1)) \Bigr| \\
    & + \sum_{j=1}^{J-1}\Bigl|  {\widetilde}{H}(u^{j} + v^{j}, n^{\pm j } + {\widetilde}{m}^{\pm j }) (\delta j) -{\widetilde}{H}(u^{j}, n^{\pm j } ) (\delta j) \Bigr|\\
    & + \Bigl| {\widetilde}{H}( u^1, n^{\pm 1})(0) - {H}({\operatorname{I}\!}u^1, n^{\pm 1})(0) \Bigr|, \end{aligned}$$ where ${\widetilde}{H}$ is the modified Hamiltonian given in [@Kish2 Section 3]. It is defined by $${\widetilde}{H}(u,n^\pm) = \| {\operatorname{I}\!}u \|_{\dot{H}^1}^2 + \frac12 \| n^\pm \|_{L^2}^2 + \frac12 \sum_{\sum k_j =0} {\widehat}{u}(k_1) {\widehat}{\overline{u}}(k_2) \Bigl[ \sigma^+(k_1,k_2) {\widehat}{n}^+(k_3) +\sigma^-(k_1,k_2) {\widehat}{n}^-(k_3) \Bigr] .$$ Here $\sigma^\pm$ is the bounded nonsingular Fourier multiplier operator defined in [@Kish2 (3.2)-(3.4)]. The effect of the multiplier $\sigma^\pm$ is to eliminate certain low-order terms from $\frac{{\,\operatorname{d}\!}}{{\,\operatorname{d}\!}t} {\widehat}{H}(u,n^\pm)$, resulting in a more favorable growth bound for the modified Hamiltonian. The precise definition of $\sigma^\pm$ is rather lengthy (and not used here) so we omit it.

We proceed by bounding the growth of each term in the sum above. By [@Kish2 Prop. 3.2], for $s > \frac12$, we can bound the difference between the Hamiltonian energy and the modified Hamiltonian by $$\begin{gathered}
    \Bigl| H({\operatorname{I}\!}u^J, n^{\pm J})(\delta J) - {\widetilde}{H}( u^J, n^{\pm J})(\delta J) \Bigr| 
     + \Bigl| {\widetilde}{H}( u^1, n^{\pm 1})(0) - {H}({\operatorname{I}\!}u^1, n^{\pm 1})(0) \Bigr| \\
     \lesssim {N_{\text{I}}}^{-1+}  \Bigl( \| 