$\arg(x) - \pi/2 \geq -{\mathrm{Re}}(x)$.[  ]{}

Now consider the arguments of $e^{i{\theta}t}$ and $c$ from equations and : $$\begin{aligned}
{\theta}t = \arg(e^{i {\theta}t}) &= \pm \tfrac{\pi}{2} \pm \tfrac{1}{2{\theta}^2} + O(\tfrac{1}{{\theta}^4}), \\ \vspace{4pt}
\arg(c) &= \pm\tfrac{\pi}{2} \pm \tfrac{1}{2{\theta}} + O(\tfrac{1}{{\theta}^2}), \\ \vspace{4pt}
\arg(ce^{i2{\theta}^2 t}) &= \arg(c) + 2{\theta}({\theta}t) \\ \vspace{4pt}
& = \pm \tfrac{\pi}{2}  \pm \tfrac{1}{2{\theta}}[2 \pm 1] +O(\tfrac{1}{{\theta}^2}). \vspace{4pt}\end{aligned}$$ On the other hand from we have ${\mathrm{Re}}(ce^{i2{\theta}^2 t}) = -2/\sqrt{n} \approx  -\tfrac{1}{{\theta}}$, and so $$\arg(ce^{i2{\theta}^2 t}) = \pm \tfrac{\pi}{2}  - \tfrac{1}{{\theta}} +O(\tfrac{1}{{\theta}^2}),$$ a contradiction. Using Lemma \[lem:argbound\] this asymptotic argument can be made exact. Details are left to the reader.

The characterizing equations of the complement reduce to $$\begin{aligned}
ce^{i(2{\theta}^2-2)t} \sqrt{n} & = -2 \pm i\sqrt{4{\theta}^2-5} \label{eqn:flatsrg2.1c} \\
ce^{i(-{\theta}-1) t} \sqrt{n} & = 1 - {\theta}+ y(-{\theta}-1) \label{eqn:flatsrg2.2c} \\
ce^{i({\theta}-1) t} \sqrt{n} & = 1 + {\theta}+ y({\theta}-1) \label{eqn:flatsrg2.3c}.\end{aligned}$$ Adding/subtracting the last two equations gives $\sqrt{n}ce^{-it}{\mathrm{Re}}(e^{i{\theta}t}) = 1-y$ and $\sqrt{n}ce^{-it}{\mathrm{Im}}(e^{i{\theta}t}) = {\theta}(1+y)$. Again we can solve: $$\begin{aligned}
e^{i{\theta}t} & = \frac{1}{\sqrt{n({\theta}^2-1)}}[\pm 1 \pm {\theta}i\sqrt{4{\theta}^2-5}] \\
ce^{-it} & = \frac{1}{2\sqrt{{\theta}^2-1}}[\pm 1 \pm i\sqrt{4{\theta}^2-5}].\end{aligned}$$ Again we use an asymptotic description of the argument which can be made precise with Lemma \[lem:argbound\]. We get: $$\begin{aligned}
{\theta}t = \arg(e^{i {\theta}t}) &= \pm \tfrac{\pi}{2} \pm 