remains to the left of the peak of $p_-$; $$\norm{u(t)}_\h{1}=\dist(\f(t),\f_c) < \ep$$ This holds for $T_0< t<T_2$, $T_0< T$ so we have a contradiction and conclude $\dist(\f(t),\f_c)<\ep$ for all $t< T$.

Now we relax $\calN[\f_0] = \calN[\f_c]$. First apply Lemma \[lem:nearby-wave\] to $\f_c$, to find $\delta_c$ that will define the $\h{1}$ neighborhood about the origin where the initial perturbation must reside. Let $K$ be the constant such that $$\norm{\f_c - \f_{c'}}_\h{1} \leq \tilde K \abs{c-c'}\leq \tilde K K_c \norm{u}_\h{1}=K \norm{u}_\h{1},$$ where $K_c$ is from Lemma \[lem:nearby-wave\]. We will seek a new wave $\f_{c'}$ with which to apply the preceding argument; however, $\ep_\star$ and the polynomials $p_\pm$ will be determined by $\f_{c'}$. But $\f_{c'}$ is determined by $\f_0$ and we have not yet found *all* the bounds $\f_0$ must satisfy. Uniform control is needed. Using $\mathcal{G}$, the implicit function associated with Lemma \[lem:nearby-wave\], let $$\begin{aligned}
c_{\min} &=& \min_{\norm{u}_\h{1}<\delta_c} \mathcal{G}[u]\\
c_{\max} &=& \max_{\norm{u}_\h{1}<\delta_c} \mathcal{G}[u]\end{aligned}$$ and let $$\ep_\star' = \min_{c\in [c_{\min},c_{\max}]} \ep_\star(\f_c)$$ Similarly, the coefficients of $p_\pm$ may be chosen such that $$p'_-\paren{\norm{u}_\h{1}} \leq \abs{\calE[\f_c+u]-\calE[\f_c]}=\abs{\Delta \calE} \leq p'_+\paren{\norm{u}_\h{1}}$$ for all $c \in [c_{\min},c_{\max}]$, $\norm{u}_\h{1} < \min\{\delta_c,\frac{1}{2}\}$ (the $\frac{1}{2}$ is required by Lemma \[lem:energybound\]).

Given $\f_c$, let $\ep_\star'$ be as above. Let $\ep_\star = 