a minimum of $Q$ at some time $t$, we have $g g_{ss} - Ag_s^2 - Bf_s^2 \le -\bar{C}$ for a large constant $\bar{C}>0$ to be chosen. Because we are bounding $Q$ from below, we may assume that $g_{ss}\leq0$. Then since Corollary \[cor-gs\] and Lemma \[lem-fs\] give uniform bounds for $|f_s|$ and $|g_s|$, we may choose $\bar{C}$ sufficiently large relative to $A$ and $B$ such that $$\label{eq-help-term}
-g g_{ss} \left(\frac{4f^2}{g^4} + \frac{2f_s^2}{f^2} + \frac{4Ag_s^2}{g^2}\right) \ge \frac{\bar{C} f^2}{2 g^4}
+ \frac{\bar{C} f_s^2}{2f^2} + \frac{\bar{C} A g_s^2}{g^2}.$$ It then follows from that at a minimum of $Q$ at time $t$, we have $$\label{eq-Q100}
\begin{split}
\frac{d}{dt} Q_{\min} &\ge 2Bf_{ss}^2 +  \frac{\bar{C} f^2}{2 g^4}
+ \frac{\bar{C} f_s^2}{2f^2} + \frac{\bar{C} A g_s^2}{g^2} \\
&- \frac{16 B f^3 f_s g_s}{g^5} - \frac{24 f f_s g_s}{g^3} - \frac{8A f f_s g_s}{g^3} + \frac{2g f_s^3 g_s}{f^3}
- \frac{8 g_s^2}{g^2} - \frac{8A g_s^2}{g^2}\\ 
&+ \frac{4 f f_{ss}}{g^2} + \frac{4B f_s^2 f_{ss}}{f} - \frac{8B f_s g_s f_{ss}}{g} - \frac{2g f_s g_s f_{ss}}{f^2}.
\end{split}$$ To estimate the terms in containing $f_{ss}$, we use Lemma \[NearCalabi\], Corollary \[cor-gs\], Lemma \[lem-fs\], the facts that $f \le g$ and $|g_s|\leq f/g$, and a weighted Cauchy–Schwarz inequality to determine that there exists a uniform constant $C'$ 