about profiles with constant scale. Thanks to Lemma \[lemme allure de la solution\] and obvious boundaries from below we get for any ${\displaystyle}{\tau < {\displaystyle}{ \tau^{*}_{j_{0}} \eqdefa \inf_{j \in \mathcal{J}_{1}}{{T_{*}}(\varphi^{j})}} =1 }$ $$\begin{split}
\| NS(v_{0,n})(\tau) \|^2_{\dot{H}^s} \geqslant \sum_{j \in \mathcal{J}_{1}} \| NS(\varphi^{j})(\tau) \|^{2}_{\dot{H}^s} - |\gamma_{n}^{J}(\tau)|.\\ 
\end{split}$$ Among profiles with a scale equal to $1$ (e.g $j \in \mathcal{J}_{1}$), we distinguish profiles with a lifespan equal to $\tau^{*}_{j_{0}} =1$ and profiles with a lifespan $\tau^{*}_{j}$ strictly greater than $1$. In other words, we consider the set $$\tilde{\mathcal{J}_{1}} \eqdefa \{ j \in \mathcal{J}_{1} \, \, | \,\, \tau^{*}_{j} = 1 \}.$$ Therefore, for any $\tau < 1$, $$\begin{split}
\| NS(v_{0,n})(\tau) \|^2_{\dot{H}^s} &\geqslant \| NS(\varphi^{j_{0}})(\tau) \|^2_{\dot{H}^s} + \sum_{j \in \tilde{\mathcal{J}_{1}}, \, j \neq j_{0}} \, \| NS(\varphi^{j})(\tau) \|^{2}_{\dot{H}^s}\\  &+ \sum_{j \in \mathcal{J}_{1} \setminus \tilde{\mathcal{J}_{1}}} \,\| NS(\varphi^{j})(\tau) \|^{2}_{\dot{H}^s}\, - |\gamma_{n}^{J}(\tau)|, 
\end{split}$$ which be bounded from below once again by $$\label{equation de reference}
\begin{split}
\| NS(v_{0,n})(\tau) \|^2_{\dot{H}^s} \geqslant \| NS(\varphi^{j_{0}})(\tau) \|^2_{\dot{H}^s} + \sum_{j \in \tilde{\mathcal{J}_{1}}, \, j \neq j_{0}} \, \| NS(\varphi^{j})(\tau) \|^{2}_{\dot{H}^s}\, - |\gamma_{n}^{J}(\tau)|,
\end{split}$$ since obviously the term ${\displaystyle}{\sum_{j \in \mathcal{J}_{1} \setminus \tilde{\mathcal{J}_{1}}} \,\| NS(\varphi^{j})(\tau) \|^{2}_{\dot{H}^s}\,}$ is positive.\
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$\bullet$ Step $4$ : in order to complete the proof of Lemma \[general lemma 