We prove that Smale's mean value constant $K^*$, defined below, exceeds $1.1559837$. The best lower bound proved before was
$1.1018$, due to Jatar and Ng~\cite{JatarNg2026b}. We also prove other results on the pairs of a univalent map and a finite Blaschke
product through which Jatar and Ng recast the problem. After the first version of this paper, the author proved that
$K^*=4$~\cite[Thm.~1.1]{GoelK4}.

\subsection*{Smale's mean value problem}
Let $P$ be a complex polynomial of degree $n\ge2$ and let $z$ be a point with $P'(z)\neq0$. Smale~\cite{Smale1981} proved that some
critical point $\zeta$ of $P$ satisfies
\[
  \Bigl|\frac{P(z)-P(\zeta)}{z-\zeta}\Bigr|\le K\,|P'(z)|
\]
with $K=4$, and he asked for the best constant. After a normalisation (Section~\ref{sec:extremal}) the best constant in degree $n$ is
$K(n)=\sup_{p\in\mathcal P_n}m(p)$, where $\mathcal P_n$ is the set of polynomials of degree $n$ with $p(0)=0$ and $p'(0)=1$, and
$m(p)$ is the minimum of $|p(b)/b|$ over the critical points $b$ of $p$. We write $K^*=\sup_nK(n)$ for Smale's mean value constant.
The polynomial $z-z^n/n$ gives $K(n)\ge(n-1)/n$, and Smale asked whether $K^*=1$, or
even $K(n)=(n-1)/n$ (Smale's (1981) mean value conjecture); in his list of problems for the next century~\cite{Smale2000} it is the first of the problems in the
addendum.

Every known improvement of $4$ is a bound for $K(n)$ that depends on $n$. The conjecture holds for $n\le4$ (Tischler~\cite{Tischler1989}; see also~\cite[Sect.~7.2]{RahmanSchmeisser2002}),
and Crane announced a computer-assisted proof for $n=5$~\cite[Sect.~1]{Crane2007}.
Beardon, Minda and Ng~\cite{BeardonMindaNg2002} proved $K(n)\le4^{(n-2)/(n-1)}$, Conte, Fujikawa and
Lakic~\cite{ConteFujikawaLakic2007} proved $K(n)\le4(n-1)/(n+1)$, Fujikawa and Sugawa~\cite{FujikawaSugawa2006} combined the two ideas,
and Crane~\cite{Crane2007} proved $K(n)<4-2.263/\sqrt n$ for $n\ge8$. All these bounds tend to $4$ as $n\to\infty$, so none of them improves the bound $K^*\le4$ for the
constant itself; in fact $K^*=4$~\cite{GoelK4}. Crane~\cite{Crane2006} showed that in every degree there is an extremal polynomial
all of whose critical points attain the minimum in $m(p)$, and $K(n)$ is nondecreasing in $n$ (see~\cite{Crane2006}
and~\cite[proof of Cor.~4.6]{JatarNg2026b}). For the dual problem see~\cite{DubininSugawa2009,JatarNg2026a}.

\subsection*{Recent results}
In September 2026 a proof that $K^*>1$ was made public~\cite{LeanMVP2026}; it shows that some $p$ has $m(p)>1$, but does
not make the degree of $p$ explicit. So Smale's conjecture is false. Adamczewski and Colbrook~\cite{AdamczewskiColbrook2026} explain the mechanism behind that proof and give an
explicit polynomial $p$ of degree $29\,984$ with $m(p)>1.0000004$, verified by interval arithmetic, so $K(n)>1$ for every $n\ge29\,984$.
In~\cite{JatarNg2026b} Jatar and Ng recast the
problem in univalent function theory. A translation of the problem into one about finite Blaschke products goes back to
Sheil-Small~\cite[Ch.~10]{SheilSmall2002}; see also Ng and Zhang~\cite{NgZhang2016} and Dubinin~\cite{Dubinin2024}.
Jatar and Ng proved that
\[
  K^*=\sup\Bigl\{\frac1{B(0)}\Bigr\},
\]
the supremum over pairs $(W,B)$ of a normalised univalent map $W$ of the unit disk $\mathbb D$ (that is, $W\in\mathcal S$: $W$ is
holomorphic and injective, $W(0)=0$, $W'(0)=1$) and a finite Blaschke product $B$ with
$B(0)>0$ such that the derivative of the product $WB$ has no zero in $\mathbb D$~\cite[Thm.~1.2]{JatarNg2026b}; Jatar and Ng call such pairs \emph{admissible}\label{term:admissible}. They also proved that $K^*$ equals the supremum
$\Lambda_{\rm JN}$ of $\min_{\partial\Omega}|\Phi(z)/z|$ over bounded analytic Jordan domains $\Omega\ni0$ and $\Phi$ locally univalent on a
neighbourhood of $\overline\Omega$ with $\Phi(0)=0$, $\Phi'(0)=1$~\cite[Thm.~4.4]{JatarNg2026b}; we call this the \emph{domain formulation}, and $\min_{\partial\Omega}|\Phi(z)/z|$ the \emph{ratio} of the pair. Writing $\mathcal B_d$\label{term:Bd} for the supremum over $\deg B\le d$, they proved
\[
  1.1018586788\ldots=\sqrt{8081/6656}\le\mathcal B_1<\sqrt{7/5}=1.1832159566\ldots,
\]
so that $K^*>1.1018$~\cite[Thm.~1.3]{JatarNg2026b}, and
$\mathcal B_d<4-\frac{529}{576}\frac{(\log2)^2}{d^2}$ for every $d\ge1$. For a second explicit pair, $\mathcal B_1>1.1030$ is stated without proof
in~\cite[Rem.~6.7]{JatarNg2026b}; Corollary~\ref{cor:ES2} proves it. They also give a winding criterion for admissibility, show that every degree of $B$
occurs and that each $\mathcal B_d$ is attained, and show that a starlike $W$ carries no nonconstant $B$. Jatar and Ng~\cite[Sect.~9]{JatarNg2026b}
ask whether $\mathcal B_d$ is strictly increasing, whether $\mathcal B_1=K^*$, and whether
$K^*=4$. The last two questions are now answered: $K^*=4$~\cite{GoelK4}, and so $\mathcal B_1<K^*$ by the upper bound
for $\mathcal B_1$ displayed above.

\subsection*{The main result}
\begin{theorem}\label{thm:intro}
Let $J_{\rm I}$ be the rational number
\[
  J_{\rm I}=\frac{2679525219362263370274816001747955444675871769085742015997}{2005184811347697816912979513414954793585480256604979855360}.
\]
Then $K^*\ge\sqrt {J_{\rm I}}=1.15598372919\ldots$, so $K^*>1.1559837$, and $K(n)>1.1559837$ for all large $n$.
\end{theorem}

\begin{figure}[ht]
\centering
\definecolor{openK}{RGB}{34,139,34}%
\definecolor{shutK}{gray}{0.86}%
\begin{tikzpicture}[x=3.6cm,y=0.85cm,font=\small]
  \fill[openK] (1,3.3) rectangle (4,3.8);
  \draw (1,3.3) rectangle (4,3.8);
  \node[left,inner sep=9pt] at (1,3.55) {Smale 1981~\cite{Smale1981}};
  \node[text=white] at (2.55,3.55) {$1\le K^*\le4$};
  \filldraw[fill=white,draw=black,thick] (1,3.55) circle (4pt);
  \fill[shutK] (1,2.2) rectangle (1.1018586,2.7);
  \fill[openK] (1.1018586,2.2) rectangle (4,2.7);
  \draw (1,2.2) rectangle (4,2.7);
  \node[left,inner sep=9pt] at (1,2.45) {Jatar and Ng 2026~\cite{JatarNg2026b}};
  \node[text=white] at (2.55,2.45) {$1.1018<K^*\le4$};
  \fill[shutK] (1,1.1) rectangle (1.1559837,1.6);
  \fill[openK] (1.1559837,1.1) rectangle (4,1.6);
  \draw (1,1.1) rectangle (4,1.6);
  \node[left,inner sep=9pt,align=right] at (1,1.35) {\textbf{This paper}\\(Theorem~\ref{thm:intro})};
  \node[text=white] at (2.55,1.35) {$1.1559<K^*\le4$};
  \fill[shutK] (1,0) rectangle (4,0.5);
  \draw (1,0) rectangle (4,0.5);
  \node[left,inner sep=9pt] at (1,0.25) {Later~\cite{GoelK4}};
  \node at (2.55,0.25) {$K^*=4$};
  \filldraw[fill=openK,draw=black,thick] (4,0.25) circle (5pt);
  \draw (1,-0.35) -- (4,-0.35);
  \foreach \k in {1,2,3,4} {\draw (\k,-0.28) -- (\k,-0.42) node[below] {$\k$};}
\end{tikzpicture}
\caption{What is known about $K^*$, to scale. In each row the green part is where $K^*$ may still lie; the grey part is ruled out.
Smale proved $K^*\le4$ in 1981 and conjectured $K^*=1$ (white dot); that conjecture was disproved in
2026~\cite{LeanMVP2026,AdamczewskiColbrook2026}. After the first version of this paper, the author proved that $K^*=4$~\cite{GoelK4}
(green dot).}\label{fig:known}
\end{figure}

Here $J_{\rm I}=1/B(0)^2$ for the explicit admissible pair of Theorem~\ref{thm:twozeroK2N18}.\footnote{Theorem~\ref{thm:intro} is proved in Lean~4 from Lean's three standard axioms alone, with no other result encoded as an axiom, and checked with the Lean comparator: \url{https://github.com/david44hello/papers_supporting_materials/tree/smale-mean-value-constant-v2.90/smale-mean-value-constant}} Theorem~\ref{thm:intro} is
Theorem~\ref{thm:twozeroK2N18} and Corollary~\ref{cor:Kd}; Figure~\ref{fig:known} shows the known range before and after it.
The bound comes from one explicit admissible pair, with $B$ of
degree two, built from a polynomial $Q$ of degree $18$ whose derivative has a zero of order $5$ on the circle, by a construction of
Jatar and Ng~\cite[Lemma~6.2]{JatarNg2026b} that we extend from one zero of $B$ to several (Lemma~\ref{lem:kzero}). The required
properties are checked in exact arithmetic (Lemma~\ref{lem:certify} and Appendix~\ref{app:pairK2N18}). Besides the cited results of Jatar and Ng, Dieudonn\'e and Crane, the proof of Theorem~\ref{thm:intro} needs only
Section~\ref{sec:prelim}, Lemma~\ref{lem:SC}, Section~\ref{sec:onezero} up to Theorem~\ref{thm:twozeroK2N18},
Corollary~\ref{cor:Kd} and Appendix~\ref{app:pairK2N18}. The proof of Theorem~\ref{thm:intro} alone is also given in~\cite{GoelShort}.

\subsection*{Other results}
The other results are about the pairs themselves. A \emph{Blaschke model} is a pair $(W,B)$ of a map $W\in\mathcal S$ that is
univalent on a neighbourhood of the closed disk and a finite Blaschke product $B$ such that $(WB)'$ has no zero on the closed disk
(Definition~\ref{def:model}); after $B$ is multiplied by a unimodular constant, it is admissible. The \emph{gain} of a pair $(W,B)$ is $1/|B(0)|$, and the gains of Blaschke models have supremum $K^*$ (Theorem~\ref{thm:BF}). A pair is \emph{extremal}\label{term:extremal} if it
attains some $\mathcal B_d$ with $d\ge1$. Items (1)--(3) are the main results after
Theorem~\ref{thm:intro}. Table~\ref{tab:results} compares the best or closest earlier results with ours. To our
knowledge, the results stated without a source are new. Section~\ref{sec:open} collects open problems.

\begin{enumerate}
\item[(1)] \emph{Degrees one and two} (Section~\ref{sec:onezero}). $1.1363375<\mathcal B_1<1.177801$ and
  $1.1559837<\mathcal B_2<100/61<1.639345$. The lower bounds come from explicit pairs (Theorems~\ref{thm:onezeroN18}
  and~\ref{thm:twozeroK2N18}); the upper bounds come from the homotopy argument of~\cite[Thm.~7.4]{JatarNg2026b}, run on other
  circles and checked in interval arithmetic (Theorems~\ref{thm:B1upper} and~\ref{thm:B2upper}).
\item[(2)] \emph{Every degree} (Section~\ref{sec:DEG}). $\mathcal B_d\le4-0.00597/(d\log d)$ for every $d\ge2$
  (Theorem~\ref{thm:DEG}), by Dubinin's separating transformation~\cite{Dubinin1990}; this improves~\cite[Thm.~1.3]{JatarNg2026b} for $d\ge452$.
\item[(3)] \emph{Extremal pairs do not extend holomorphically past the circle} (Sections~\ref{sec:ES}--\ref{sec:ratext}). The map $W$ of an extremal pair does not
  extend holomorphically to any neighbourhood of the closed disk (Corollary~\ref{cor:S}), and it is not rational
  (Corollary~\ref{cor:ratext}). In particular no extremal pair is a Blaschke model (Corollary~\ref{cor:ES}).
\item[(4)] \emph{Counting from the circle} (Section~\ref{sec:count}). If $W$ is meromorphic on a neighbourhood of the closed disk,
  the zeros of $(WB)'$ in the disk can be counted from the circle, and every admissible pair has $\deg B\le\operatorname{wt}(W)$, a
  number read off $W$ on the circle (Theorem~\ref{thm:count}).
\item[(5)] \emph{Pockets} (Section~\ref{sec:pockets}). For a Blaschke model, a \emph{pocket} is a point $e^{i\theta}$ of the circle
  where $\frac{d}{d\theta}|W(e^{i\theta})|=0$ and $\frac{d}{d\theta}\arg W(e^{i\theta})<0$ (Definition~\ref{def:pocket}). At a pocket the \emph{depth} is $x=-\bigl(\frac{d}{d\theta}\arg W(e^{i\theta})\bigr)^{-1}>0$; let
  $\sigma=\sum_jP_{a_j}$ be the sum of the Poisson kernels $P_{a_j}(w)=(1-|a_j|^2)/|w-a_j|^2$ of the zeros $a_j$ of $B$. The
  winding number in the criterion of~\cite[Sect.~8]{JatarNg2026b} is a signed count of the pockets with $\sigma x>1$, when no
  pocket has $\sigma x=1$ (Theorem~\ref{thm:pocket}).
\item[(6)] \emph{Ownership} (Section~\ref{sec:own}). Each zero $a$ of $B$ in a Blaschke model \emph{owns} a pocket at a local minimum of $|W|$: there $\sigma x>1$,
  but $(\sigma-P_a)x\le1$. Distinct zeros own distinct pockets (Theorem~\ref{thm:own}).
\item[(7)] \emph{Gains tending to $4$} (Section~\ref{sec:structure}). Let $(W_n,B_n)$ be admissible pairs with gains tending to $4$.
  After rotations, $W_n$ tends to the Koebe function
  and $B_n$ tends to $(1-w)^2/4$ (Theorem~\ref{thm:C}); the zeros $a$ of $B_n$, weighted by $\log(1/|a|)$, have an explicit limit
  distribution (Proposition~\ref{prop:green}).
\item[(8)] \emph{Symmetric pairs} (Section~\ref{sec:symmetric}). Every $n$-fold symmetric Blaschke model comes from the $n$-th root
  transform of~\cite[Rem.~8.6]{JatarNg2026b} (Theorem~\ref{thm:rtconverse}). The transform keeps the gain, but it gives a Blaschke
  model only under an explicit condition (Lemma~\ref{lem:rt}).
\item[(9)] \emph{Zeros of $B$ are not small} (Section~\ref{sec:FH}). Every zero of $B$ in a Blaschke model has modulus greater than
  $0.36967$ (Corollary~\ref{cor:FH}).
\item[(10)] \emph{Fixed degree} (Section~\ref{sec:fixeddeg}). Take a pair of the construction of~\cite[Lemma~6.2]{JatarNg2026b} built
  from a polynomial of degree at most $n$. If it has fewer than $n$ boundary defects (zeros of $(WB)'$, critical points of $W$, and
  pairs of points where the curve $W(e^{i\theta})$ meets itself, all on the circle and counted with weights), then a nearby pair of
  the same kind has a larger gain (Theorem~\ref{thm:fixeddeg}).
\item[(11)] \emph{Adding zeros near the circle} (Section~\ref{sec:addzeros}). If $W(w)\not\equiv w$ is meromorphic on a neighbourhood of the closed disk, we
  count exactly the zeros of $(WB)'$ in the disk after zeros of $B$ are added on fixed radii near the circle, with $W$ kept; and we show that one zero of $B$ added near the circle on the radius through a zero of
  $(WB)'$ on the circle makes the pair inadmissible (Proposition~\ref{prop:BC}).
\item[(12)] \emph{Other exact pairs, and Dubinin's conjecture} (Section~\ref{sec:onezero}). Table~\ref{tab:onezero} lists eighteen
  pairs with exact rational data (Theorem~\ref{thm:tablepairs}). Remark~\ref{rem:Dubinin} notes that, by~\cite{LeanMVP2026} and~\cite{Crane2006}, Dubinin's
  conjecture~\cite[(1.5)]{Dubinin2024}, which would give $K(n)=(n-1)/n$, fails in every large degree, and it
  shows how a polynomial $p$ of degree $n$ with $m(p)>(n-1)/n$ gives Blaschke products that violate it.
\end{enumerate}

{\small
\begin{longtable}{@{}p{0.48\textwidth}@{\hspace{0.03\textwidth}}p{0.49\textwidth}@{}}
\caption{The best or closest result known before this paper, next to each new result.}\label{tab:results}\\
\toprule
\textbf{Best result before this paper} & \textbf{New result}\\
\midrule
\endfirsthead
\toprule
\textbf{Best result before this paper} (continued) & \textbf{New result}\\
\midrule
\endhead
$K^*>1.1018$~\cite[Thm.~1.3]{JatarNg2026b}; $K^*>1.1030$, stated without
  proof in~\cite[Rem.~6.7]{JatarNg2026b} & $K^*>1.1559837$, and $K(n)>1.1559837$ for all large $n$\\[2pt]
$1.1018<\mathcal B_1<\sqrt{7/5}=1.1832159\ldots$~\cite[Thm.~1.3]{JatarNg2026b}; $\mathcal B_1>1.1030$~\cite[Rem.~6.7]{JatarNg2026b} & $1.1363375<\mathcal B_1<1.177801$\\[2pt]
$\mathcal B_2<3.8897$~\cite[Thm.~1.3]{JatarNg2026b} & $1.1559837<\mathcal B_2<100/61<1.639345$\\[2pt]
$\mathcal B_d<4-529(\log2)^2/(576d^2)$~\cite[Thm.~1.3]{JatarNg2026b};
  the new bound is better for $d\ge452$ & $\mathcal B_d\le4-0.00597/(d\log d)$ for every $d\ge2$\\[2pt]
\cite[Rem.~8.11]{JatarNg2026b} left open whether a pair that attains $\mathcal B_d$ can be regular
  (a Blaschke model) & If $(W,B)$ attains $\mathcal B_d$ with $d\ge1$, then $W$ does not extend holomorphically to any neighbourhood of the closed disk
  and is not rational\\[2pt]
--- & Every admissible pair whose $W$ is meromorphic on a neighbourhood of the closed disk has $\deg B\le\operatorname{wt}(W)$, a
  number read off $W$ on the circle\\[2pt]
after rotations $W_n$ tends to the Koebe function, all zeros of $B_n$ tend to the circle and $\deg B_n\to\infty$~\cite[Rem.~7.3]{JatarNg2026b}; such pairs exist if $K^*=4$, a question of~\cite[Sect.~9]{JatarNg2026b} that is now answered:
  $K^*=4$~\cite{GoelK4} & If admissible pairs $(W_n,B_n)$ have $1/B_n(0)\to4$, then after rotations the zeros $a$ of $B_n$, weighted by $\log(1/|a|)$, have an explicit limit
  distribution\\[2pt]
--- & A local maximiser of the gain among the pairs built from a polynomial of degree at most $n$ has at least $n$ weighted boundary
  defects\\[2pt]
the winding criterion of~\cite[Sect.~8]{JatarNg2026b} & A pocket formula for the winding number of $(WB)'/(W'B)$ along the circle, and the ownership of
  pockets by the zeros of $B$\\[2pt]
every zero has $|a|\ge1/3$~\cite[Thm.~7.1]{JatarNg2026b} & Every zero of $B$ in a Blaschke model has modulus greater than $0.36967$\\[2pt]
the transform, and this correspondence for polynomials $w\,p(w^n)$, in~\cite[Rem.~8.6]{JatarNg2026b} & Every $n$-fold symmetric pair with $B(0)\neq0$ whose map is univalent past the circle comes from the $n$-th root transform\\
\bottomrule
\end{longtable}}

\subsection*{Supporting files} The supporting archive holds the exact data of the eighteen pairs of Table~\ref{tab:onezero} and
the programs, with their outputs, for the computer steps of
Corollaries~\ref{cor:FH} and~\ref{cor:ES2}, Section~\ref{sec:flatmax},
Theorems~\ref{thm:onezeroN18}, \ref{thm:twozeroK2N18}, \ref{thm:tablepairs}, \ref{thm:B1upper} and~\ref{thm:B2upper},
and Corollary~\ref{cor:K2N18BM}. It is deposited at
\begin{center}\url{https://doi.org/10.5281/zenodo.23597020}\end{center}

\subsection*{Terms}
The terms below are introduced in this paper or used in a special sense; the last column gives the page of the definition.

{\small
\begin{longtable}{@{}p{0.27\textwidth}@{\hspace{0.02\textwidth}}p{0.61\textwidth}@{\hspace{0.02\textwidth}}r@{}}
\toprule
\textbf{Term} & \textbf{Meaning} & \textbf{p.}\\
\midrule
\endhead
admissible pair $(W,B)$ & $W\in\mathcal S$, $B$ a finite Blaschke product with $B(0)>0$, and $(WB)'\neq0$ in $\mathbb D$ & \pageref{term:admissible}\\
$\mathcal B_d$ & the supremum of $1/B(0)$ over admissible pairs with $\deg B\le d$ & \pageref{term:Bd}\\
gain & $1/|B(0)|$ & \pageref{def:model}\\
extremal pair & an admissible pair that attains some $\mathcal B_d$ with $d\ge1$ & \pageref{term:extremal}\\
$m(p)$, $K(n)$, $K^*$ & the constants of Smale's problem & \pageref{def:K}\\
domain pair, ratio, $\Lambda$, $\Lambda_{\rm JN}$ & $(\Omega,\Phi)$ of the domain formulation; $\min_{\partial\Omega}|\Phi(z)/z|$; its supremum over all domain pairs, resp.\ over those of~\cite[Thm.~4.4]{JatarNg2026b} & \pageref{def:admissible}\\
Blaschke model & a pair with (M1)--(M3): $W$ univalent past the circle, $(W B)'\neq0$ on the closed disk & \pageref{def:model}\\
regular pair & Jatar and Ng's name for a Blaschke model with $B(0)>0$ & \pageref{term:regular}\\
triple; $\sigma$, $v$, $\gamma$ & $(W,B,w_0)$ with a base point $w_0$; on the circle $\sigma=wB'/B$, $v=(W-W(w_0))/(wW')$,
  $\gamma=1+\sigma v$ & \pageref{term:triple}\\
feasible & $((W-W(w_0))B)'$ has no zero on the closed disk &
  \pageref{term:feasible}\\
back-facing point & a boundary point $\zeta$ of $W(\mathbb D)$ at which $\operatorname{Re}v<0$ & \pageref{term:backfacing}\\
pocket; depth, sign & a point of the circle where $v<0$; $x=-v$, and the sign change $\varepsilon$ of $\operatorname{Im}v$ there &
  \pageref{def:pocket}\\
on, off (pocket) & $\sigma x>1$, resp.\ $\sigma x<1$ & \pageref{def:pocket}\\
frame; base-$0$ depth & the choice of the base point; the depth $x_0$ seen from base point $0$ & \pageref{term:frame}\\
$K_0$ (Sections~\ref{sec:structure} and~\ref{sec:permodel}) & the gain $1/|B(0)|$ of a pair normalised by $B(0)>0$ & \pageref{term:model}\\
$\Omega$, $A$, $\Phi$ (Sections~\ref{sec:structure}--\ref{sec:FH}) & $W(\mathbb D)$; the zero set of $B$; $\Phi(z)=zB(W^{-1}(z))$ on $\Omega$, so that $\Phi\circ W=WB$ & \pageref{term:F4}\\
$E_b$ (Sections~\ref{sec:structure}--\ref{sec:FH}) & the component of $\{|\Phi|<r_{\rm in}\}$ that contains the zero $b$ of $\Phi$ & \pageref{lem:branch}\\
exit point & a boundary point of $W(\mathbb D)$ where an inverse branch of $\Phi=zB(W^{-1}(z))$ reaches the boundary &
  \pageref{lem:exit}\\
owned point & a point of the circle that $W$ maps to an exit point of the branch at $W(a_j)$ & \pageref{term:owned}\\
$n$-fold symmetric pair & $W(e^{2\pi i/n}w)=e^{2\pi i/n}W(w)$ and the zeros of $B$ are invariant under $w\mapsto e^{2\pi i/n}w$ & \pageref{term:symmetric}\\
$F$ (from Section~\ref{sec:ES} on) & $F=WB$, so that $F=\Phi\circ W$ & \pageref{sec:ES}\\
$w$, $z$ & $W$, $B$ and $F$ have variable $w$ throughout; for the pairs built from a polynomial, $Q$ and $H$ have variable $z=\varphi(w)$ (Sections~\ref{sec:onezero}, \ref{sec:fixeddeg}, and where Sections~\ref{sec:permodel} and~\ref{sec:addzeros} use these pairs); $z$ also denotes points of $\Omega=W(\mathbb D)$ & \pageref{sec:onezero}\\
$\mathrm{Crit}$, $\mathrm{Cont}$; contact pair & critical points of $W$ on $\mathbb T$; points $\zeta\neq\zeta'$ of $\mathbb T$
  with $W(\zeta)=W(\zeta')$ & \pageref{term:crit}\\
slit plane & $\mathbb C$ minus a non-empty finite union of analytic arcs, closed in $\mathbb C$ & \pageref{term:slit}\\
$Z_-$ & the points of $\mathbb T$ where $wW'/W<0$: $|W|$ is stationary and $\arg W$ decreases & \pageref{term:Zminus}\\
index $\iota_\zeta$; $Z_\chi$ & the contribution of a point $\zeta$ of $\mathbb T$ with $wF'/F\le0$ to the count of zeros of $F'$ &
  \pageref{term:index}\\
$\operatorname{wt}(W)$ & a sum over $Z_-$, read off $W$; it bounds $\deg B$ & \pageref{term:wt}\\
flat maximum & a point of $Z_-$ where $\log|W|$ has a strict local maximum and its derivative vanishes to order at least $3$ &
  \pageref{term:flatmax}\\
$\mathcal L_n$, $\mathcal G_n$ & pairs of the construction of~\cite[Lemma~6.2]{JatarNg2026b} with polynomials of degree at most $n$
  & \pageref{term:Ln}\\
boundary defects; $n_H$ & zeros of $(WB)'$, critical points and contact pairs of $W$ on $\mathbb T$; $n_H-2$ counts them with weights &
  \pageref{term:defects}\\
margin & the distance from the unit circle to the nearest zero of a polynomial & \pageref{term:margin}\\
$\mathcal D_c$ & the Dieudonn\'e polynomials of a polynomial $H$, $-1\le c\le1$ & \pageref{lem:dieudonne}\\
\mbox{$\chi$-negative}, \mbox{$\chi$-positive};\newline \mbox{$\chi$-depth} & a zero $x$ of $\frac d{d\theta}\log|W|$ on $\mathbb T$ with $\chi(x)<0$, resp.\
  $\chi(x)>0$, where $\chi=wF'/F$; $\eta(x)=-\chi(x)$ & \pageref{term:chineg}\\
hook & a slit that turns back toward $0$ & \pageref{term:hook}\\
other letters & a letter used only inside one proof, remark or computation is defined there; $[z^j]P$ is the coefficient of $z^j$
  in a polynomial $P$ & \\
\bottomrule
\end{longtable}}
\addtocounter{table}{-1}%