\section{The data of pair \texorpdfstring{\ref{pair:N18}}{VI}}\label{app:pairN18}

The zeros are
\[
  a=\tfrac{5104807}{16777216}+\tfrac{5629749}{16777216}\,i,\qquad b=\tfrac{7625713}{16777216}-\tfrac{10960539}{16777216}\,i .
\]
Then
\[
  Q'(z)=18\,(z-1)^5\,(z-\rho_0)\prod_{j=1}^{11}(z-\rho_j),
\]
with
\begin{align*}
\rho_1 &= \tfrac{255391}{524288}+\tfrac{1143071}{1048576}\,i, &
\rho_2 &= \tfrac{88671}{1048576}+\tfrac{1246607}{1048576}\,i, &
\rho_3 &= -\tfrac{58043}{262144}+\tfrac{1155163}{1048576}\,i,\\
\rho_4 &= -\tfrac{150829}{262144}+\tfrac{480931}{524288}\,i, &
\rho_5 &= -\tfrac{892479}{1048576}+\tfrac{334139}{524288}\,i, &
\rho_6 &= -\tfrac{751281}{1048576}-\tfrac{841275}{1048576}\,i,\\
\rho_7 &= -\tfrac{477637}{1048576}-\tfrac{1061955}{1048576}\,i, &
\rho_8 &= -\tfrac{90629}{1048576}-\tfrac{1208029}{1048576}\,i, &
\rho_9 &= \tfrac{255145}{1048576}-\tfrac{1211027}{1048576}\,i,\\
\rho_{10} &= \tfrac{18039}{32768}-\tfrac{1032877}{1048576}\,i, &
\rho_{11} &= \tfrac{949373}{1048576}-\tfrac{669111}{1048576}\,i .
\end{align*}
Put $Q(z)=\int_b^zQ'(t)\,dt$, so that $Q(b)=0$ and $Q$ is monic of degree $18$. The root $\rho_0$ is fixed by $Q(a)=0$: with
$\mathcal V(t)=(t-1)^5\prod_j(t-\rho_j)$,
\[
  \rho_0=\frac{\int_b^a t\,\mathcal V(t)\,dt}{\int_b^a \mathcal V(t)\,dt},
\]
a Gaussian rational whose real and imaginary parts have denominators of $234$ digits; numerically
$\rho_0=0.85175723\ldots+0.82891526\ldots\,i$. Then $H(z)=(1-\bar az)\,Q(z)/(z-a)$ is a polynomial of degree $18$, and
$R(z)=18\,(z-\rho_0)\prod_j(z-\rho_j)$.

For these data the facts (D), (C), (R) and (S) of the proof of Theorem~\ref{thm:onezeroN18} hold; each is verified in exact arithmetic over
$\mathbb Q(i)$. For (C) the division of $Q'$ by $(z-1)^5$ is exact, $R(1)\neq0$, and Lemma~\ref{lem:SC} is applied to $R$, of
degree $12$; for (S) it is applied to $H'$, of degree $17$. (R) is checked as described after Lemma~\ref{lem:certify}, with the resultant of degree at most $578$ in $c$ computed at the
$581$ integer points $c=0,1,\dots,580$.
The margins are small. The zero of $R$ nearest to the circle has modulus $1.06329949\ldots$. The zero of $H'$ nearest to the circle
has modulus $1.00001014\ldots$; since $\mathcal D_{\pm1}(w)=H'(\pm w)$, this is the margin at $c=\pm1$. Numerically the zeros of the $\mathcal D_c$
come slightly closer near $c=\pm0.3257$, within about $9.24\cdot10^{-6}$. Condition (S) concerns $c=\pm1$ only; the other $c$ are covered by (R).