predetermined constant.

We will further decompose $w_0$ as a sum $$\begin{aligned}
 \label{datum:0}
w_0=\underbrace{\kappa_0+\eps^{\frac14} {\overline}{W_2}\left(\eps^{-\frac54}\theta\right) \chi(\eps^{-1}\theta) + \eps^{\frac 1 4} {\widehat}{W}_0(\eps^{- \frac 5 4} \theta)}_{=: \check w_0(\theta)}+ \eps^{\frac 1 4} \Big(\alpha (\eps^{- \frac 5 4} \theta)^2+\beta (\eps^{- \frac 5 4} \theta)^3 \Big) \chi(\eps^{- \frac 5 4} \theta)\,. \end{aligned}$$ for some smooth fixed cut-off, $\chi$, satisfying $\chi(x) = 1$ for $|x| \le 1$ and is supported in a ball of radius $2$. Above the constants $\alpha, \beta$ are determined by ${\widehat}W_0$ and are not free parameters that we choose as part of the data. The perturbation ${\widehat}W_0$ will be assumed to satisfy the following $$\begin{aligned}
{\left \| {\widehat}W_0 \right\|}_{C^8\left(\left[ -\frac{M\eps}{2} ,\frac{M\eps}{2}\right]\right)} &\leq \eps^2\label{eq:C8:bnd}
\\
{\widehat}W_0^{(n)}(0)&=0,\quad\mbox{for } n=0,1,4, 5\label{eq:W0:diff}\\ \label{est:hatW:in}
{\left|{\widehat}W_0^{(n)}(0)\right|}&\leq \eps,\quad\mbox{for } n=2, 3 \,.\end{aligned}$$ We also assume the following bounds on $z$ and $a$ $$\begin{aligned}
{\left \| z_0 \right\|}_{C^8} + {\left \| a_0 \right\|}_{C^8} &\leq \eps^2\,.\end{aligned}$$

Main theorem
------------

We now state our main theorem:

\[thm:general\] Let $\gamma>1$ be given and set $ \lambda = {\tfrac{\gamma-1}{2}}$. Then there exists a sufficiently large $\kappa_0 = \kappa_0(\lambda) > 0$, sufficiently large $M = M(\lambda,\kappa_0) \geq 1$, and sufficiently small $\eps = \eps(\lambda,\kappa_0,M) \in (0,1)$ such that the following holds:

Let $(w_0,z_0,a_0)$ be initial 