the principal branch of SOPS does not fold back on itself, and (ii) there are no other connected components (*isolas*) of SOPS. By [@BergJaquette; @lessard2010recent; @jlm2016Floquet; @xie1991thesis] the principal branch does not have any folds $ \alpha > {\tfrac{\pi}{2}}$.

In [@jlm2016Floquet; @xie1991thesis] it is shown that there is a unique SOPS for $ \alpha \geq 1.9$. These proofs use that fact that if every SOPS is asymptotically stable for some $ \alpha > {\tfrac{\pi}{2}}$, then there is a unique SOPS [@xie1993uniqueness]. Using estimates describing SOPS for when $\alpha$ is large [@nussbaum1982asymptotic], Xie showed that there is a unique SOPS for all $ \alpha \geq 5.67$ [@xie1991thesis]. By using computer-assisted proofs to characterize SOPS to Wright’s equation [@jlm2016Floquet], this method was extended to show there is a unique SOPS for $ \alpha \in [1.9,6.0]$.

However, for $ \alpha$ close to the bifurcation value $ {\tfrac{\pi}{2}}$ the dynamics becomes center-like, and proving uniqueness through these stability arguments becomes infeasible. To overcome this obstacle, we equate the problem of finding periodic orbits to with a zero–finding problem in a space of Fourier coefficients. We then employ rigorous numerics to derive a computer-assisted proof that there is a unique SOPS to Wright’s equation for 