Let $A \gtrsim 1$. Let $J \subset [0,\infty)$. Let $w_{l}$ be a solution of (\[Eqn:NlkgWdat\]). Let $(C,Q,M) \in \mathbb{R}^{+} \times
\mathbb{R}^{+} \times[100, \infty)$. There exist $\eta_{1}:=\eta_{1}(p)>0$, $\alpha:=\alpha(p)>0$, $\bar{\alpha}:=\bar{\alpha}(p)$ and $C_{1}:=C_{1}(p)>> 1$ such that if

$$\begin{array}{l}
\sup_{t \in J} E(Iw_{l}(t))  \leq \left( C_{E} N^{2(1-s)} A^{p+1} - l (1 - \epsilon_{0}) (N^{s-1} \eta_{1} A^{-1})^{\alpha} \right) \left( 1 +
\beta \right)
\end{array}
\label{Eqn:InducNrj}$$

$$\begin{array}{l}
\| I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}(J)} \leq 2 X_{C,Q}
\end{array}
\label{Eqn:InducStr}$$

and

$$\begin{array}{ll}
\|  I w_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+} (J)} & \geq e^{ (C_{1} M e^{ C_{1} (N^{1-s} A)^{\alpha}} ) ^{ C_{1} (N^{1-s} A )^{\alpha}}}
\end{array}$$

then there exists $T_{l} \in J$ and $v_{l}$, solution of the free Klein-Gordon equation, that satisfy

$$\begin{array}{l}
E(Iw_{l}- Iv_{l},T_{l}) \leq  C_{E} N^{2(1-s)} A^{p+1} - (l+1)(1- \epsilon_{0})(N^{s-1}  \eta_{1} A^{-1})^{\alpha}
\end{array}
\label{Eqn:SepMolNrj}$$

$$\begin{array}{l}
E_{c}(v_{l},T_{l}) \lesssim (N^{s-1} \eta_{1}  A^{-1})^{\alpha}
\end{array}$$

and

$$\begin{array}{l}
\| I v_{l} \|_{L_{t}^{2(p-1)-} L_{x}^{2(p-1)+}(T_{l},b)} \lesssim \frac{(N^{1-s} \eta_{1} A)^{\alpha}}{M^{\bar{\alpha}}}
\end{array}
\label{Eqn:FreeDecay}$$

assuming that $(N,s)$ satisfies the following conditions

$$\begin{array}{ll}
N^{1-s} A^{\frac{p+1}{2}} \frac{( A^{-1} \eta_{1} N^{s-1})^{(p-1) \alpha} }{ N^{\frac{5-p}{2}-}} & = o \left( N^{s-1} \eta_{1} A^{-1}
\right)^{\alpha}
\end{array}
\label{Eqn:Cond1}$$

$$\begin{array}{ll}
\frac{( N^{1-s} A^{\frac{p+1}{2}} )^{p-1} }{N^{\frac{5-p}{2}-}} & << 1
\end{array}
\label{Eqn:Cond2}$$

and

$$\begin{array}{ll}
\frac{(X_{C,Q})^{2(p-1)-} (N^{1-s} A^{\frac{p+1}{2}})^{p} }{N^{\frac{5-p}{2}-}} & << 1
\end{array}
\label{Eqn:Cond3}$$

Here

$$\begin{array}{ll}
\beta & := \frac{ 