V \oplus (T^*M\otimes V),\\
 {\mathbf{Y}}(\tau) &= {\operatorname{A}}(\tau) \oplus {\operatorname{B}}(\tau) \in   (T^*M\otimes V)\oplus (T^2(T^*M)\otimes V)\end{aligned}$$ where $V = {\operatorname{End}}({\wedge^2 T^*M})$. The families ${\mathbf{X}}$ and ${\mathbf{Y}}$ of sections are smoothly defined on all of $E_{R_0}\times[0, 1]$ and vanish identically on $E_{R_0}\times \{0\}$.

Combining with Proposition \[prop:sysev\], we see that $$\begin{aligned}
\label{eq:xysys}
\begin{split}
    \left|{\frac{\partial {\mathbf{X}}}{\partial\tau}} + \Delta{\mathbf{X}}\right| &\leq \frac{C}{r^2_c}\left(|{\mathbf{X}}| + |{\mathbf{Y}}|\right)\\
    \left|{\frac{\partial {\mathbf{Y}}}{\partial\tau}}\right| &\leq C\left(|{\mathbf{X}}| + |\nabla {\mathbf{X}}|\right) + \frac{C}{r_c^2} |{\mathbf{Y}}|
\end{split}\end{aligned}$$ for some $C = C(n, K_0)$ on $E_{R_0}\times [0, 1]$. Here $|\cdot| = |\cdot|_{g(\tau)}$, $\nabla = \nabla_{g(\tau)}$, and $\Delta =\Delta_{g(\tau)}$. Moreover, we also have $$\sup_{E_{R_0}\times[0, 1]}\{|{\mathbf{X}}| + |\nabla {\mathbf{X}}| + |{\mathbf{Y}}|\} \leq C(n, K_0).$$ The boundedness of the components of ${\mathbf{X}}$ follows directly from . For the remaining components, note first that we can bound ${{\frac{\partial }{\partial\tau}}}\nabla {\hat{{\operatorname{P}}}}$ via , and hence also ${\operatorname{A}}= \nabla {\hat{{\operatorname{P}}}}$. With , we can then bound the components of $\nabla{\mathbf{X}}$. Similarly, one can estimate ${\operatorname{B}}= \nabla\nabla {\hat{{\operatorname{P}}}}$ via the equation for ${{\frac{\partial }{\partial\tau}}}\nabla\nabla {\hat{{\operatorname{P}}}}$ and the bound on $\nabla{\hat{{\operatorname{P}}}}$.

The vanishing of ${\mathbf{X}}$ and ${\mathbf{Y}}$ near infinity is now a consequence of the Carleman estimates established in Propositions 4.7, 5.7, and 5.9 of [@KotschwarWangConical].

\[thm:bu\] Suppose $(M, g(\tau))$ is a self-similar solution to on $E_{R_0}\times 