data satisfying the assumptions stipulated in Section \[ss:initial\], with the constants $\alpha$ and $\beta$ are left to be chosen. Then, there exists $\alpha$, $\beta$ satisfying ${\left|\alpha\right|}+{\left|\beta\right|}\leq \eps^{\frac{9}{10}}$ and a corresponding solution $(a,z,w) \in C([-\eps,T_*); C^8({{\mathbb T}}))$ to satisfying the following properties:

-   The solution forms a singularity at a computable time $T_*$ and angle $\theta_*$.

-   $\sup_{t\in[-\eps, T_*)} \left( \| a\|_{ W^{1, \infty }({{\mathbb T}})} + \| z\|_{ W^{1, \infty} ({{\mathbb T}})} + \| w\|_{ L^\infty({{\mathbb T}})} \right) \leq  C_M,$

-   $\lim_{t \to T_*}   {\ensuremath{\partial}}_\theta w(\xi(t),t) = -\infty $ and $\frac{1}{2(T_*-t)} \leq  {\left \| {\ensuremath{\partial}}_\theta w(\cdot,t) \right\|}_{L^\infty} \leq \frac{2}{T_*-t}$ as $t \to T_*$,

-   $w( \cdot , T_*)$ has a cusp singularity of Hölder $C^ {\sfrac{1}{5}} $ regularity

Moreover, $w$ blows up in an asymptotically self-similar manner described by the profile ${\overline}{W}_2$. Specifically, there exists a $\nu>0$ and $\kappa_*$ such that $$\begin{aligned}
\lim_{t\rightarrow 0} (-t)^{-\frac1{4}}\left(w(x(-t)^{-\frac{5}{4}},t)-\kappa_*\right)= \left(\frac{\nu}{120}\right)^{-\frac14}{\overline}{W}_2\left(\left(\frac{\nu}{120}\right)^{\frac14}x\right)\,,\end{aligned}$$ where $\nu=\lim_{t \to T_*}   (T_*-t)^{-8}w^{(5)}(\xi(t),t)$ and $\kappa_*=\lim_{t \to T_*} \kappa(t) $ are explicitly computable, satisfying ${\left|\nu-120\right|}\leq \eps^{\frac 3 4}$ and ${\left|\kappa_0-\kappa_*\right|}\leq \eps$.

As a corollary, we show that Theorem is stable modulo a two dimensional linear subspace of initial data:

\[c:open\] There exists an open set $\Xi$ of initial data $(\check w_0,z_0,a_0)$ in the 