\[prop:BifNbd\] apply.

\[prop:zeroneighborhood2\] Let $\omega \geq 1.1$, $\alpha \in (0,2]$, and define $$g(\alpha,\omega) := \sqrt{ \left(1- \tfrac{\omega}{\alpha} \right)^2 + 2   \, \tfrac{\omega}{\alpha} \left( 1- \sin  \omega \right)} .
    \label{eq:ZeroNbd}$$ If $F(\alpha,\omega,c)=0$, then either $c \equiv 0$ or $     g(\alpha,\omega) \leq  \|c \|_{\ell^1}   $.

\[prop:BifNbd\] For each $\alpha  \in  ({\tfrac{\pi}{2}}, {\tfrac{\pi}{2}}+ 0.00553] $ there is a unique (up to time translation) periodic solution to Wright’s equation with Fourier coefficients satisfying $  \|c \|_{\ell^1} \leq  0.18$ and having frequency $ | \omega - {\tfrac{\pi}{2}}| \leq 0.0924$.

We note that while Lemma \[prop:zeroneighborhood2\] is stated only for $ \omega \geq 1.1$ and $ \alpha \in (0,2]$, a more general formula is given in [@BergJaquette]. Also, we present the hypothesis of Lemma \[prop:BifNbd\] in terms of a bound on $ \| c\|_{\ell^1}$ as opposed to a bound on $ \| y'\|_{L^2}$ as in the original paper. This allows us to use the stronger result derived in the proof of [@BergJaquette Theorem 4.10], namely that the solution *exists* and is unique, as opposed to the exact result stated in [@BergJaquette Theorem 4.10], which is that there is most one periodic solution.

The remainder of this section is dedicated to proving Lemma \[prop:CentralLemma\], which estimates $F_\infty$, 