\section{The constants in degrees one and two}\label{sec:onezero}
\newcounter{onezeropair}\renewcommand{\theonezeropair}{\Roman{onezeropair}}

Jatar and Ng~\cite[Lemma~6.2]{JatarNg2026b} build pairs with one zero from a polynomial. Let $Q$ be a polynomial with two distinct
zeros $a,b\in\mathbb D$ and with $Q'\neq0$ in $\mathbb D$, and suppose that the polynomial
\[
  H(z)=\frac{(1-\bar a z)\,Q(z)}{z-a}
\]
is univalent in $\mathbb D$. Put $\varphi(w)=(w+b)/(1+\bar bw)$, $N_b=(1-|b|^2)H'(b)$, $W=(H\circ\varphi)/N_b$ and let $B$
be the multiple of $B_a\circ\varphi$ by the constant of modulus $1$ that makes $B(0)>0$. Then $(W,B)$ is admissible, and, as $1/|B(0)|=|1-\bar ab|/|a-b|$ and
$|1-\bar ab|^2-|a-b|^2=(1-|a|^2)(1-|b|^2)$,
\[
  \frac1{|B(0)|^2}=J(a,b):=1+\frac{(1-|a|^2)(1-|b|^2)}{|a-b|^2}.
\]
With $\deg Q=4$ they obtain $\mathcal B_1\ge\sqrt{8081/6656}=1.1018\ldots$~\cite[Thm.~6.1]{JatarNg2026b}. We use the same
construction, extended to two zeros (Lemma~\ref{lem:kzero}), and Lemma~\ref{lem:certify} gives a
test in exact arithmetic; the pairs of Table~\ref{tab:onezero} have $6\le\deg Q\le18$. With one zero and $\deg Q=18$ we obtain an admissible pair with gain larger than $1.1363375$
(Theorem~\ref{thm:onezeroN18}), and with two zeros and $\deg Q=18$ one with gain larger than $1.1559837$
(Theorem~\ref{thm:twozeroK2N18}). Both gains are attained exactly by Blaschke models as well (Corollaries~\ref{cor:recordBM}
and~\ref{cor:K2N18BM}). Theorem~\ref{thm:tablepairs} and Table~\ref{tab:onezero} give the other exact pairs.

For the univalence of a polynomial $H$ of degree at most $n$, with coefficient $[z^j]H$ of $z^j$, we use the Dieudonn\'e polynomials of $H-H(0)$,
\[
  \mathcal D_c(w)=\sum_{j=1}^n [z^j]H\,U_{j-1}(c)\,w^{j-1}\qquad(c\in[-1,1]),
\]
where $U_{j-1}$ is the Chebyshev polynomial of the second kind. For $c=\cos\theta$ they satisfy
$H(ze^{i\theta})-H(ze^{-i\theta})=2i\sin\theta\;z\mathcal D_c(z)$, and $\mathcal D_1(w)=H'(w)$, $\mathcal D_{-1}(w)=H'(-w)$.

\begin{lemma}[{Dieudonn\'e's criterion~\cite{Dieudonne1931}; see also~\cite[Lemma~6.3]{JatarNg2026b}}]\label{lem:dcrit}
$H$ is univalent in $\mathbb D$ if no $\mathcal D_c$ with $c\in[-1,1]$ has a zero in $\mathbb D$.
\end{lemma}

\begin{lemma}\label{lem:dieudonne}
Suppose that no $\mathcal D_c$ with $c\in[-1,1]$ has a zero on the circle, and that $H'$ has no zero on the closed disk. Then $H$ is
univalent on a disk $\{|z|<r\}$ with $r>1$.
\end{lemma}

\begin{proof}
$\mathcal D_c(0)=H'(0)\neq0$, so no $\mathcal D_c$ vanishes identically. As no $\mathcal D_c$ has a zero on the circle, the number
$\frac1{2\pi i}\oint_{|w|=1}\mathcal D_c'/\mathcal D_c\,dw$ of zeros of $\mathcal D_c$ in $\mathbb D$ is a continuous integer-valued function of $c\in[-1,1]$, hence
constant; it is $0$ at $c=1$, where $\mathcal D_1=H'$. So no $\mathcal D_c$ has a zero on the closed disk. The set of pairs $(c,w)$ with $c\in[-1,1]$,
$|w|\le2$ and $\mathcal D_c(w)=0$ is compact, so there is $r\in(1,2)$ such that no $\mathcal D_c$ vanishes on $\{|w|\le r\}$. The Dieudonn\'e polynomials
of $H(rz)/r$ are $\mathcal D_c(rw)$, so $H(rz)/r$ is univalent in $\mathbb D$ by Lemma~\ref{lem:dcrit}.
\end{proof}

\begin{corollary}\label{cor:ES2}
The maps $W$ of the degree-one pairs of~\cite[Thm.~6.1]{JatarNg2026b} and~\cite[Rem.~6.7]{JatarNg2026b} are univalent on a
neighbourhood of the closed disk, so these pairs are not extremal. The pair of~\cite[Rem.~6.7]{JatarNg2026b} is admissible, with
gain $\sqrt{1768591361/1453520000}=1.10307\ldots$; this proves the bound $\mathcal B_1>1.1030$ stated in~\cite[Rem.~6.7]{JatarNg2026b}.
\end{corollary}

\begin{proof}
For both pairs $Q=(z-1)^4-u^4$ with $u=-x-iy$ \cite[proof of Thm.~6.1]{JatarNg2026b}, where $(x,y)=(9/16,3/8)$ for~\cite[Thm.~6.1]{JatarNg2026b}
and $(x,y)=(14/25,3/8)$ for~\cite[Rem.~6.7]{JatarNg2026b}; so $Q'=4(z-1)^3$ has no zero in $\mathbb D$.
For each pair let $H$ be the quartic of~\cite[Sect.~6]{JatarNg2026b}, with $b$, $\varphi$ and $N_b$ as at the start of this section, and $\mathcal D_c$ the Dieudonn\'e polynomial of $H-H(0)$, of formal degree
$3$ with coefficients depending on $c\in[-1,1]$. Its three quantities $\delta_1,\delta_2,\delta_3$ of Lemma~\ref{lem:SC} are real
polynomials in $c$ of degrees $6$, $12$ and $24$; an exact computation with Sturm sequences shows that they have no zero in $[-1,1]$ and
are positive at $c=0$. By Lemma~\ref{lem:SC} no $\mathcal D_c$ has a zero in the closed disk, so by compactness there is $r\in(1,1/|b|)$ such
that no $\mathcal D_c$ vanishes on $\{|w|\le r\}$. The Dieudonn\'e polynomials of $H(rz)/r$ are $\mathcal D_c(rw)$, so $H$ is univalent on $\{|z|<r\}$,
and $W=H(\varphi)/N_b$ \cite[Lemma~6.2]{JatarNg2026b} is univalent on a neighbourhood of the closed disk. As $Q'\neq0$ in $\mathbb D$,
both pairs are admissible by~\cite[Lemma~6.2]{JatarNg2026b}; for the pair of~\cite[Rem.~6.7]{JatarNg2026b},
$1/B(0)^2=1768591361/1453520000$ exactly, by~\cite[(15)]{JatarNg2026b} with $(x,y)=(14/25,3/8)$. Lemma~\ref{lem:ES} applies with $k=1$.
\end{proof}

The construction of~\cite[Lemma~6.2]{JatarNg2026b} extends to several zeros.

\begin{lemma}\label{lem:kzero}
Let $k\ge1$ and let $Q$ be a polynomial with $Q'\neq0$ in $\mathbb D$ and with $k+1$ distinct zeros $b,a_1,\dots,a_k\in\mathbb D$,
and suppose that the polynomial
\[
  H(z)=Q(z)\prod_{j=1}^k\frac{1-\bar a_jz}{z-a_j}
\]
is univalent in $\mathbb D$. With $\varphi$ and $N_b=(1-|b|^2)H'(b)$ as above, put $\eta=\prod_jB_{a_j}(b)$, $W=(H\circ\varphi)/N_b$ and
$B=(|\eta|/\eta)\prod_jB_{a_j}\circ\varphi$. Then $(W,B)$ is admissible, $\deg B=k$, and
\[
  \frac1{B(0)^2}=\prod_{j=1}^kJ(a_j,b).
\]
\end{lemma}

\begin{proof}
$H$ is a polynomial since the $a_j$ are distinct zeros of $Q$, and $H(b)=0$ since $Q(b)=0$ and $b$ is no $a_j$. As $H$ is univalent,
$H'(b)\neq0$; as $\varphi$ is an automorphism of $\mathbb D$, $W$ is univalent in $\mathbb D$, with $W(0)=H(b)/N_b=0$ and
$W'(0)=H'(b)\varphi'(0)/N_b=1$. Each $B_{a_j}\circ\varphi$ is a Blaschke factor, so $B$ is a Blaschke product of degree $k$, and
$B(0)=|\eta|>0$. Since $H\prod_jB_{a_j}=Q$, we have $(WB)'=(|\eta|/(\eta N_b))\,Q'(\varphi)\,\varphi'$, which has no zero in $\mathbb D$.
Finally $1/|B_{a_j}(b)|^2=J(a_j,b)$, by the identity at the start of this section.
\end{proof}

\begin{lemma}\label{lem:recipek}
Let $Q$, $b,a_1,\dots,a_k$, $H$, $W$ and $B$ be as in Lemma~\ref{lem:kzero}. Suppose that $Q'$ has no zero on the closed disk
and that $H$ is univalent on a disk $\{|z|<r\}$ with $r>1$. Then $(W,B)$ is a Blaschke model with $k$ zeros and gain
$\prod_{j=1}^kJ(a_j,b)^{1/2}$.
\end{lemma}

\begin{proof}
$\varphi$ is holomorphic and injective on $\{|w|<1/|b|\}$ and maps the closed disk onto itself, so by continuity it maps a disk
$\{|w|<\varrho\}$ with $1<\varrho<1/|b|$ into $\{|z|<r\}$, and $W$ is univalent there. With $W(0)=0$ and $W'(0)=1$ from the proof of
Lemma~\ref{lem:kzero}, this is (M1). (M2) is clear. By the proof of Lemma~\ref{lem:kzero}, $(WB)'=(|\eta|/(\eta N_b))\,Q'(\varphi)\,\varphi'$;
it has no zero on the closed disk, since $Q'$ has none there and $\varphi$ maps the closed disk onto itself. This is (M3). The gain is
$1/B(0)=\prod_jJ(a_j,b)^{1/2}$ by Lemma~\ref{lem:kzero}.
\end{proof}

Lemma~\ref{lem:certify} reduces admissibility to four conditions that exact arithmetic can verify.

\begin{lemma}\label{lem:certify}
Let $k\ge1$, let $Q$ be a polynomial of degree $n$, let $b,a_1,\dots,a_k\in\mathbb C$, put
$H(z)=Q(z)\prod_{j=1}^k(1-\bar a_jz)/(z-a_j)$, and let $\mathcal D_c$ be the Dieudonn\'e polynomials of $H-H(0)$. Suppose that the following
hold.
\begin{enumerate}
\item[(D)] $|b|,|a_1|,\dots,|a_k|<1$, the $k+1$ points are pairwise distinct, and $Q(b)=Q(a_1)=\dots=Q(a_k)=0$.
\item[(C)] $Q'=\Pi R$, where $R$ has no zero on the closed disk, and $\Pi$ is either a non-zero constant or a constant times
  $(z-\zeta_1)^{m_1}\cdots(z-\zeta_p)^{m_p}$ with $p\ge1$, distinct $\zeta_i$, $|\zeta_i|=1$ and $m_i\ge1$.
\item[(R)] Let $\mathcal D_c^*(w)=w^{n-1}\overline{\mathcal D_c(1/\bar w)}$, at formal degree $n-1$. The resultant $\operatorname{Res}_w(\mathcal D_c,\mathcal D_c^*)$, taken
  at formal degrees, is a polynomial in $c$ of degree at most $2(n-1)^2$, and it has no zero in $[-1,1]$.
\item[(S)] $H'$ has no zero on the closed disk.
\end{enumerate}
Then $H$ is univalent on a disk $\{|z|<r\}$ with $r>1$; the pair $(W,B)$ of Lemma~\ref{lem:kzero} is admissible, with $\deg B=k$
and $1/B(0)^2=\hat J:=\prod_{j=1}^kJ(a_j,b)$; $W$ is univalent on a disk $\{|w|<\varrho\}$ with $\varrho>1$; and if $\Pi$ is constant, $(W,B)$ is a
Blaschke model. Hence $\mathcal B_k\ge\sqrt{\hat J}$ and $K^*\ge\sqrt{\hat J}$.
\end{lemma}

\begin{proof}
By (D), $H$ is a polynomial of degree at most $n$, so each $\mathcal D_c$ has formal degree $n-1$. The Sylvester matrix of
$\mathcal D_c$ and $\mathcal D_c^*$ has size $2(n-1)$, and each entry is $0$, $[z^j]H\,U_{j-1}(c)$ or $\overline{[z^j]H}\,U_{j-1}(c)$ for
some $j\le n$, of degree at most $n-1$ in $c$; so the resultant is a polynomial in $c$ of degree at most $2(n-1)^2$. Let $c\in[-1,1]$ and suppose that
$\mathcal D_c(w_0)=0$ for some $w_0$ with $|w_0|=1$. Since $1/\bar w_0=w_0$, we get $\mathcal D_c^*(w_0)=w_0^{n-1}\overline{\mathcal D_c(w_0)}=0$, so $w_0$ is a
common zero of $\mathcal D_c$ and $\mathcal D_c^*$, and the resultant vanishes at $c$ (a common zero makes the Sylvester matrix singular, at any
formal degrees). This contradicts (R). So no $\mathcal D_c$ with $c\in[-1,1]$ has a zero on the circle, and by (S) and Lemma~\ref{lem:dieudonne} $H$ is univalent on a disk $\{|z|<r\}$ with $r>1$. By (C),
$Q'\neq0$ in $\mathbb D$, so Lemma~\ref{lem:kzero} applies: $(W,B)$ is admissible, with $\deg B=k$ and $1/B(0)^2=\hat J$. As in the proof
of Lemma~\ref{lem:recipek}, $W$ is univalent on a disk $\{|w|<\varrho\}$ with $\varrho>1$. If $\Pi$ is constant, $Q'$ has no zero on the closed
disk, and $(W,B)$ is a Blaschke model by Lemma~\ref{lem:recipek}. So $\mathcal B_k\ge\sqrt{\hat J}$, and $K^*\ge\sqrt{\hat J}$
by~\cite[Prop.~4.2]{JatarNg2026b}, which holds for every admissible pair.
\end{proof}

For $k=1$, Lemma~\ref{lem:kzero} is~\cite[Lemma~6.2]{JatarNg2026b}, with $B=(|\eta|/\eta)\,B_a\circ\varphi$. In the theorems below, (C)
and (S) are checked by Lemma~\ref{lem:SC}, and (R) as follows. The resultant is computed exactly (from residues modulo primes, with a
Hadamard bound) at the $2(n-1)^2+3$ integer points $c=0,1,\dots,2(n-1)^2+2$; its coefficients are obtained by exact
interpolation through the first $2(n-1)^2+1$ values and checked at the other two. Its imaginary part
vanishes at all of these points, and its degree is at most $2(n-1)^2$, so it has real
coefficients; Descartes' rule of signs with bisection then shows that it has no zero in $[-1,1]$. With one zero and $\deg Q=18$ we obtain the following pair.

\begin{theorem}\label{thm:onezeroN18}
Let $a$, $b$ and $Q$ be the data of pair~\ref{pair:N18} in Appendix~\ref{app:pairN18}, and let $(W,B)$ be the pair of
Lemma~\ref{lem:kzero} with $k=1$ and $a_1=a$. Then $(W,B)$ is an admissible pair with one
zero and $1/B(0)=\sqrt{J_{\rm VI}}$, where
\[
  J_{\rm VI}=J(a,b)=\frac{2842976349398406707505693241}{2201702138784778082456698880},\qquad \sqrt{J_{\rm VI}}=1.13633750601\ldots
\]
Hence $\mathcal B_1\ge\sqrt{J_{\rm VI}}$ and $K^*\ge\sqrt{J_{\rm VI}}>1.1363375$.
\end{theorem}

\begin{proof}
The facts (D), (C) with $\Pi=(z-1)^5$ and $R(1)\neq0$, (R) and (S) of Lemma~\ref{lem:certify}, with $k=1$ and $n=18$ (so in (R)
formal degree $17$ and degree at most $578$ in $c$), are checked in exact rational arithmetic over $\mathbb Q(i)$.
Lemma~\ref{lem:certify} gives the rest.
\end{proof}

Here $(WB)'$ has one zero on the circle, of order $5$, at $\varphi^{-1}(1)$, so pair~\ref{pair:N18} is not regular. Since $W$ is
univalent on a neighbourhood of the closed disk, Lemma~\ref{lem:ES} (or Corollary~\ref{cor:S2}) shows that pair~\ref{pair:N18} is not
extremal, so $\mathcal B_1>\sqrt{J_{\rm VI}}$. By Corollary~\ref{cor:recordBM}, Blaschke models attain this gain.

A larger gain comes from a pair with two zeros, with $\deg Q=18$ and $Q'$ of the same shape.

\begin{theorem}\label{thm:twozeroK2N18}
Let $b$, $a_1$, $a_2$ and $Q$ be the data of pair~\ref{pair:K2N18} in Appendix~\ref{app:pairK2N18}, and let $(W,B)$ be the pair of
Lemma~\ref{lem:kzero} with $k=2$. Then $(W,B)$ is an admissible pair with two zeros and $1/B(0)=\sqrt{J_{\rm I}}$, where $J_{\rm I}=J(a_1,b)\,J(a_2,b)$,
\[
  J_{\rm I}=\frac{2679525219362263370274816001747955444675871769085742015997}{2005184811347697816912979513414954793585480256604979855360},
\]
and $\sqrt{J_{\rm I}}=1.15598372919\ldots$; hence $\mathcal B_2\ge\sqrt{J_{\rm I}}$ and $K^*\ge\sqrt{J_{\rm I}}>1.1559837$.
\end{theorem}

\begin{proof}
The facts (D), (C) with $\Pi=(z-1)^5$ and $R(1)\neq0$, (R) and (S) of Lemma~\ref{lem:certify}, with $k=2$ and $n=18$ (so in (R)
formal degree $17$ and degree at most $578$ in $c$), are checked in exact rational arithmetic over $\mathbb Q(i)$: (R) as described after
Lemma~\ref{lem:certify}, with $581$ integer points (as in~\cite[Sect.~5]{GoelShort}). By Lemma~\ref{lem:certify}, $(W,B)$ is
admissible, with $\deg B=2$ and $1/B(0)^2=J_{\rm I}$. Hence $\mathcal B_2\ge\sqrt{J_{\rm I}}$, and $K^*\ge\sqrt{J_{\rm I}}$ by~\cite[Prop.~4.2]{JatarNg2026b}.
\end{proof}

Here $(WB)'$ has one zero on the circle, of order $5$, at $\varphi^{-1}(1)$, so pair~\ref{pair:K2N18} is not
regular; since $W$ is univalent on a neighbourhood of the closed disk, Lemma~\ref{lem:ES} shows that pair~\ref{pair:K2N18} does not
attain $\mathcal B_2$, so $\mathcal B_2>\sqrt{J_{\rm I}}$. By
Corollary~\ref{cor:K2N18BM}, Blaschke models with two zeros attain this gain.

Theorems~\ref{thm:onezeroN18} and~\ref{thm:twozeroK2N18} use~\cite[Prop.~4.2]{JatarNg2026b}, which gives $K^*\ge1/B(0)$ for every
admissible pair $(W,B)$; its proof first replaces $(W,B)$ by the regular pairs of~\cite[Lemma~2.1]{JatarNg2026b}. Since
Corollaries~\ref{cor:recordBM} and~\ref{cor:K2N18BM} give Blaschke models with the same gains, the two bounds also follow
from~\cite[Prop.~4.3]{JatarNg2026b}, whose proof is direct: it gives $K^*\ge\mu(\Omega,\Phi)$ for every domain pair $(\Omega,\Phi)$ in
which $\Omega$ is an analytic Jordan domain, and for a Blaschke model $(W,B)$ the domain $W(\mathbb D)$ and the function
$z\mapsto zB(W^{-1}(z))/B(0)$ form such a pair, with ratio $1/|B(0)|$ since $|B|=1$ on the circle.

Lemma~\ref{lem:certify} applies in the same way to the other pairs of Table~\ref{tab:onezero}. For a pair of the table, $\Pi$ is as in the caption:
$\Pi$ is constant for the Blaschke models, pairs~\ref{pair:F12c}, \ref{pair:F8s}, \ref{pair:F8p} and~\ref{pair:F8}; $\Pi=(z-1)^m$ for the
boundary pairs other than pair~\ref{pair:OT}, with $m=5$ or $m=3$ as in the caption; and $\Pi=(z-1)^5(z-\zeta_2)$, with $\zeta_2\neq1$ on the
circle, for pair~\ref{pair:OT}.

\begin{theorem}\label{thm:tablepairs}
Let one of the pairs~\ref{pair:K2N17}--\ref{pair:K2} and~\ref{pair:N17}--\ref{pair:F8} of Table~\ref{tab:onezero} be given. Let $Q$,
$b$ and $a_1$, and also $a_2$ for pairs~\ref{pair:K2N17}--\ref{pair:K2}, be its exact data (in the supporting archive), let $k$ be the number of
the $a_j$, let $n=\deg Q$, and let $\Pi$ be as above. Then (D), (C), (R) and (S) of Lemma~\ref{lem:certify} hold. So the pair
$(W,B)$ of Lemma~\ref{lem:kzero} is admissible, with $\deg B=k$ and $1/B(0)=\sqrt{\hat J}$, where $\sqrt{\hat J}$ is at least the value in the
table ($\sqrt{\hat J}$ truncated to seven decimals); and $(W,B)$ is a Blaschke model if Table~\ref{tab:onezero} lists the pair as one.
\end{theorem}

\begin{proof}
For each of these pairs the facts are checked in exact rational arithmetic over $\mathbb Q(i)$, (C) and (S) by
Lemma~\ref{lem:SC}, and (R) as described after Lemma~\ref{lem:certify}. That lemma gives the rest.
\end{proof}

For the
boundary pairs, $(WB)'$ has zeros on the circle, at the points $\varphi^{-1}(\zeta)$ with $\Pi(\zeta)=0$, so they are admissible pairs in
the sense of~\cite{JatarNg2026b}, but not Blaschke models. For pairs~\ref{pair:N17}, \ref{pair:N16}, \ref{pair:O14},
\ref{pair:O12F}, \ref{pair:O12}, \ref{pair:T6p} and~\ref{pair:T6}, Corollary~\ref{cor:recordBM} gives Blaschke models with exactly their gains. For the other boundary pairs,
$K^*\ge\sqrt{\hat J}$ comes from~\cite[Prop.~4.2]{JatarNg2026b}, as in Lemma~\ref{lem:certify}.

The next remark relates the number of zeros of $B$ to the zeros of $(WB)'$ on the circle.

\begin{remark}[A phase count]\label{rem:phasecount}
Let $(W,B)$ be admissible with $\deg B=k$, let $F=WB$ be holomorphic near the closed disk, and let $F'$ have the zeros
$\zeta_1,\dots,\zeta_p$ on the circle, of orders $m_1,\dots,m_p$ and total order $M=m_1+\dots+m_p$. With $\chi=wF'/F$ as in
Section~\ref{sec:count}, put $\operatorname{turn}_F:=\operatorname{Re}(1+wF''/F')$ on $\mathbb T\setminus\{\zeta_1,\dots,\zeta_p\}$. In $\mathbb D$, $F$ has
the $k+1$ zeros of $W$ and $B$, all simple, and $F'$ has none; $F\neq0$ on the circle, since $|F|=|W|$ there and $W$, univalent in
$\mathbb D$ with $W(0)=0$, takes no value on the circle that it takes in $\mathbb D$. Along each arc $I$ of
$\mathbb T\setminus\{\zeta_1,\dots,\zeta_p\}$ (if $p=0$, cut the circle at any point) take a continuous branch of $\arg\chi(e^{i\theta})$. By
the argument principle the changes of these branches add up to $(M-2k)\pi$: the factor $w$ gives $2\pi$, the factor $1/F$ gives
$-2(k+1)\pi$, and each factor $(w-\zeta_i)^{m_i}$ of $F'$ gives $m_i\pi$, since $\arg(e^{i\theta}-\zeta_i)$ gains $\pi$ along
$\mathbb T\setminus\{\zeta_i\}$. Moreover
\[
  \frac{d}{d\theta}\arg\chi(e^{i\theta})=\operatorname{Re}\Bigl(1+\frac{wF''}{F'}-\frac{wF'}{F}\Bigr)=\operatorname{turn}_F-\operatorname{Re}\chi,
\]
so at a point where $\arg\chi$ takes one of the values $\pi/2+j\pi$ ($j\in\mathbb Z$), which we call \emph{levels}, its
derivative is $\operatorname{turn}_F$.

Write the limits of $\arg\chi$ at the start and at the end of $I$ as $l_I+\alpha_I\pi$ and $l'_I+\alpha'_I\pi$, with levels $l_I,l'_I$
and $\alpha_I,\alpha'_I\in[0,1)$. Near $\zeta_i$, $\chi$ is a non-zero constant times $(w-\zeta_i)^{m_i}(1+O(w-\zeta_i))$, so the two
one-sided limits of $\arg\chi$ at $\zeta_i$ differ by a multiple of $\pi$, and $\alpha$ of each arc equals $\alpha'$ of the arc before it
(for $p=0$, $\alpha=\alpha'$ since the change is $-2k\pi$).
So the total change is $\sum_I(l'_I-l_I)+\pi\sum_I(\alpha'_I-\alpha_I)=\sum_I(l'_I-l_I)=(M-2k)\pi$.
\end{remark}

\begin{proposition}\label{prop:phasecount}
In the setting of Remark~\ref{rem:phasecount}, suppose that $\operatorname{turn}_F>0$ on $\mathbb T\setminus\{\zeta_1,\dots,\zeta_p\}$.
Then $k\le M/2$.
\end{proposition}

\begin{proof}
By Remark~\ref{rem:phasecount}, $\sum_I(l'_I-l_I)=(M-2k)\pi$. On an arc $I$, once $\arg\chi$ has
reached a level $l$ it stays $\ge l$, since it can meet $l$ only with positive derivative. It cannot leave a level downwards at the
start of $I$ either: if it tends to the level $l$ there, then $u=l-\arg\chi$ satisfies $u'=-\operatorname{turn}_F+|\chi|\cos\arg\chi\le|\chi|\,|u|$, as
$\operatorname{turn}_F>0$ and $|\cos\arg\chi|=|\sin u|\le|u|$, and $|\chi|$ is bounded near the start of $I$; since $u\to0$ there, Gronwall's inequality gives $u\le0$ on
$I$. So $l'_I\ge l_I$ for every arc, and $M-2k=\sum_I(l'_I-l_I)/\pi\ge0$.
\end{proof}

For the pairs of Lemma~\ref{lem:kzero}, $Q$ has variable $z=\varphi(w)$. Then $F$ is a constant times $Q\circ\varphi$, so $M$ is the total order of the zeros of $Q'$ on the
circle, and, as $w\varphi'/\varphi=|\varphi'|$ on the circle, $\operatorname{turn}_F=|\varphi'|\,\operatorname{Re}\bigl(1+zQ''(z)/Q'(z)\bigr)$
at $z=\varphi(w)$. If $Q'(z)$ is a constant times $(z-1)^mR(z)$, with $R\neq0$ on the closed disk, and $\deg Q=n$, then
for $|z|=1$
\[
  \operatorname{Re}\Bigl(1+\frac{zQ''(z)}{Q'(z)}\Bigr)=\frac{n+1}2-\frac12\sum_{R(\xi)=0}\frac{|\xi|^2-1}{|z-\xi|^2},
\]
with the zeros $\xi$ of $R$ counted with multiplicity. So pairs~\ref{pair:K2N18}, \ref{pair:K2N17}, \ref{pair:K2N16}, \ref{pair:K2N15} and~\ref{pair:K2} of
Table~\ref{tab:onezero} have $M-2k=1$, and with one zero of $Q'$ of order $5$ on the circle a pair with $k=3$ zeros needs, by Proposition~\ref{prop:phasecount}, a point $z$ of the circle where
$\sum_{R(\xi)=0}(|\xi|^2-1)/|z-\xi|^2\ge n+1$.

\begin{table}[!ht]
\centering
\begin{tabular}{ccll}
pair & $\deg Q$ & type & $\sqrt{\hat J}\ge$\\ \hline
\refstepcounter{onezeropair}\theonezeropair\label{pair:K2N18} & 18 & boundary, two zeros & 1.1559837\\
\refstepcounter{onezeropair}\theonezeropair\label{pair:K2N17} & 17 & boundary, two zeros & 1.1518591\\
\refstepcounter{onezeropair}\theonezeropair\label{pair:K2N16} & 16 & boundary, two zeros & 1.1509994\\
\refstepcounter{onezeropair}\theonezeropair\label{pair:K2N15} & 15 & boundary, two zeros & 1.1466539\\
\refstepcounter{onezeropair}\theonezeropair\label{pair:K2}   & 14 & boundary, two zeros & 1.1392755\\
\refstepcounter{onezeropair}\theonezeropair\label{pair:N18}  & 18 & boundary       & 1.1363375\\
\refstepcounter{onezeropair}\theonezeropair\label{pair:N17}  & 17 & boundary       & 1.1361064\\
\refstepcounter{onezeropair}\theonezeropair\label{pair:N16}  & 16 & boundary       & 1.1358030\\
\refstepcounter{onezeropair}\theonezeropair\label{pair:O14}  & 14 & boundary       & 1.1346063\\
\refstepcounter{onezeropair}\theonezeropair\label{pair:OT}   & 13 & boundary       & 1.1332281\\
\refstepcounter{onezeropair}\theonezeropair\label{pair:O12F} & 12 & boundary       & 1.1315433\\
\refstepcounter{onezeropair}\theonezeropair\label{pair:O12}  & 12 & boundary       & 1.1306600\\
\refstepcounter{onezeropair}\theonezeropair\label{pair:F12c} & 12 & Blaschke model & 1.1272406\\
\refstepcounter{onezeropair}\theonezeropair\label{pair:F8s}  & 8  & Blaschke model & 1.1196072\\
\refstepcounter{onezeropair}\theonezeropair\label{pair:F8p}  & 8  & Blaschke model & 1.1158108\\
\refstepcounter{onezeropair}\theonezeropair\label{pair:T6p}  & 6  & boundary       & 1.1144943\\
\refstepcounter{onezeropair}\theonezeropair\label{pair:T6}   & 6  & boundary       & 1.1143766\\
\refstepcounter{onezeropair}\theonezeropair\label{pair:F8}   & 8  & Blaschke model & 1.1138795
\end{tabular}
\caption{Exact pairs with one zero, and pairs~\ref{pair:K2N18}, \ref{pair:K2N17}, \ref{pair:K2N16}, \ref{pair:K2N15} and~\ref{pair:K2}
with two zeros (Theorems~\ref{thm:onezeroN18}--\ref{thm:tablepairs}); $\sqrt{\hat J}$ is truncated to
seven decimals.
``Blaschke model'': $Q'$ has no zero on the closed disk.
``Boundary'': $Q'=(z-1)^mR$ exactly, with $m=5$ for pairs~\ref{pair:K2N18}, \ref{pair:K2N17}, \ref{pair:K2N16}, \ref{pair:K2N15}, \ref{pair:K2}, \ref{pair:N18}, \ref{pair:N17}, \ref{pair:N16}, \ref{pair:O14}, \ref{pair:O12F} and~\ref{pair:O12} and $m=3$ for pairs~\ref{pair:T6p} and~\ref{pair:T6},
and $Q'=(z-1)^5(z-\zeta_2)R$ with $\zeta_2\neq1$ on the circle for pair~\ref{pair:OT}; $R$ has no zero on the closed disk, so $Q'\neq0$ in $\mathbb D$.
The gains of the boundary pairs~\ref{pair:N18}, \ref{pair:N17}, \ref{pair:N16}, \ref{pair:O14}, \ref{pair:O12F}, \ref{pair:O12},
\ref{pair:T6p} and~\ref{pair:T6} are also attained by Blaschke models (Corollary~\ref{cor:recordBM}),
and that of pair~\ref{pair:K2N18} by Blaschke models with two zeros (Corollary~\ref{cor:K2N18BM}).
The exact data of all eighteen pairs are in the supporting archive.}\label{tab:onezero}
\end{table}

\begin{corollary}\label{cor:Kd}
$K(n)>1.1559837$ for all sufficiently large $n$.
\end{corollary}

\begin{proof}
Take $A=1.1559837$, which is below $K^*$ by Theorem~\ref{thm:twozeroK2N18}. Since $K^*=\sup_nK(n)$
(Definition~\ref{def:K}), some $n_0$ has $K(n_0)>A$. Since $K(n)$ is nondecreasing in $n$ (\cite{Crane2006}; see also~\cite[proof of
Cor.~4.6]{JatarNg2026b}), $K(n)>A$ for all $n\ge n_0$.
\end{proof}

The proof gives no explicit degree.

\begin{remark}[Dubinin's conjecture]\label{rem:Dubinin}
Dubinin's conjecture~\cite[(1.5)]{Dubinin2024} is the following. Let $n\ge2$ and $0<t<1$. Every Blaschke product $B$ of degree $n$ with $B(0)=0$
and $|B'(0)|=t$ has a critical point $\zeta$ with $|B(\zeta)/(\zeta B'(0))|\le f_n(t)$, where $f_n(t)<1$ is the same ratio for
$z(z^{n-1}-t)/(1-tz^{n-1})$ at its critical points in $\mathbb D$~\cite[(1.5), Lemma~1]{Dubinin2024}. He remarks that this would
imply $K(n)=(n-1)/n$. Since $K(N)>1$ for some $N$~\cite{LeanMVP2026} (see also Corollary~\ref{cor:Kd}) and $K(n)$ is nondecreasing~\cite{Crane2006}, the conjecture
fails in every degree $n\ge N$; this is the contrapositive of his remark. Take $p\in\mathcal P_n$ with
$m(p)>(n-1)/n$, write $p(z)=\alpha z\prod_i(z-a_i)$, put $B^{(R)}(z)=z\prod_i(z-a_i/R)/(1-\bar a_iz/R)$, and let $R\to\infty$. Then
$R^nB^{(R)}(z/R)\to p(z)/\alpha$ locally uniformly, so by Hurwitz's theorem the $n-1$ critical points of $B^{(R)}$ in $\mathbb D$ have
ratios tending to those of $p$, while $|(B^{(R)})'(0)|\to0$ and $f_n(t)\to(n-1)/n$ as $t\to0$~\cite[Lemma~1]{Dubinin2024}. At the
critical points outside the closed disk the ratio is at least $1/|(B^{(R)})'(0)|>1$, by Schwarz's lemma. In the same way the
inequality~\cite[(1.2)]{Dubinin2024}, which by a result of Sheil-Small~\cite{SheilSmall2002} quoted there would imply $K^*\le1$,
fails for some Blaschke product in every degree $n\ge N$. For large $n$ the ratios of $B^{(R)}$ at all its critical points can be made larger than
$1.1559837$ (Corollary~\ref{cor:Kd}).
\end{remark}