{\operatorname{I}\!}u^J(\delta J) \|_{H^1}^2 \| n^{\pm J}(\delta J) \|_{L^2} + \| {\operatorname{I}\!}u_0 \|_{H^1}^2 \| n^{\pm }_0 \|_{L^2} \Bigr) \lesssim {N_{\text{I}}}^{-1+} {N}^3.\end{gathered}$$

Next we bound the growth of the modified energy under the Zakharov flow. We use the following estimate. It is almost identical to the ${\mathbb{T}}^2$ bound [@Kish2 Prop. 4.1]. However, it is slightly stronger because of more favorable estimates available in the one-dimensional case. The proof is in Section \[prf\_of\_growth\_bound\].

\[growth\_bound\] Fix $\frac12 < s < 1$ and $\delta \in (0,1)$. Suppose $(u, n^\pm)$ is a smooth solution to on the time interval $[0,\delta]$. Then $$\begin{gathered}
    | {\widetilde}{H}(u, n^\pm)(\delta) - {\widetilde}{H}(u, n^\pm)(0)| \lesssim 
    {N_{\text{I}}}^{-1+} \delta^{\frac12 -} 
    \| {\operatorname{I}\!}u \|_{X^{1,\frac12,1}_\delta}^2 
    \| n^\pm \|_{Y^{0,\frac12,1}_{\pm\delta}} \\
    + \Bigl[ {N_{\text{I}}}^{-2+} 
    + {N_{\text{I}}}^{-\frac32+} \delta^{\frac12 -} 
    + {N_{\text{I}}}^{-1} \delta^{1-}\Bigr] 
    \left( \| {\operatorname{I}\!}u \|_{X^{1,\frac12,1}_\delta}^4  
    + \| {\operatorname{I}\!}u \|_{X^{1,\frac12,1}_\delta}^2 
    \| n^\pm \|_{Y^{0,\frac12,1}_{\pm\delta}}^2 \right).
    \end{gathered}$$

This gives $$\begin{aligned}
    \Bigl| {\widetilde}{H}( u^j, n^{\pm j})(\delta j) - {\widetilde}{H}( u^j, n^{\pm j})(\delta (j-1)) \Bigr| 
    \lesssim& 
    {N_{\text{I}}}^{-1+} \delta^{\frac12-} {N}^3  + 
    \Bigl({N_{\text{I}}}^{-2+} + {N_{\text{I}}}^{-\frac32+} \delta^{\frac12-} + {N_{\text{I}}}^{-1+} \delta^{1-}\Bigr)N^4 \\
    \approx &
    {N_{\text{I}}}^{-1+} {N}^2  + 
    {N_{\text{I}}}^{-2+}{N}^4 + {N_{\text{I}}}^{-\frac32+}{N}^{3}.\end{aligned}$$

Finally, we use the definition of the modified energy to note that $$\begin{aligned}
    \Bigl|  {\widetilde}{H}(u^{j} + &v^{j}, n^{\pm j } + {\widetilde}{m}^{\pm j }) (\delta j) -{\widetilde}{H}(u^{j}, 