In our applications, the constants $C_{L}>0$, $C_{N}>0$ and $r_{0}>0$ will be fixed as uniform constants (independent of $T\gg 1$). We will see this from the global linear and nonlinear estimates, which will be stated and proved in next subsections. With the specified weight parameters $\delta,\mu,\nu$, the error estimate in Proposition 4.23 in fact guarantees $\|\mathrm{Err}_{CY}\|_{\mathfrak{B}}\to 0$ as $T\to\infty$, which particularly implies $\|\mathscr{F}(\bm{0})\|_{\mathfrak{B}}\to 0$ and hence $\mathscr{F}$ satisfies (b) of Item (2) in the above lemma.