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  \author[Suvrit Sra]{Suvrit Sra\textsuperscript{\normalfont*}}%
  \thanks{*\,TU Munich, Dept.~of Mathematics, School of CIT; Garching, Germany.
    Email: \texttt{s.sra@tum.de}.}}

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\keywords{Macdonald polynomials, Jack polynomials, majorization, Heckman--Opdam hypergeometric function, Harish-Chandra--Itzykson--Zuber integral, Dixon--Anderson integral, Hankel positivity, Bernstein certificates}

\hypersetup{
  pdftitle={On majorization inequalities for normalized Jack and Macdonald
            polynomials},
  pdfauthor={Suvrit Sra}
}

\makeatletter
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\begin{document}
\title[On majorization for Jack and Macdonald polynomials]{On majorization inequalities for normalized\\
Jack and Macdonald polynomials}
\aiauthor
\begin{abstract}
  We study the inequality \(P_\lambda(x;q,t)/P_\lambda(1^n;q,t)\geq P_\mu(x;q,t)/P_\mu(1^n;q,t)\) for
  all \(x\in\reals_{>0}^n\) whenever the partition \(\lambda\) majorizes \(\mu\); Sra suggested it in
  2015, and Chen, Khare and Sahi stated it as a conjecture in 2025.  We prove the Jack case for every
  \(n\) from Sun's positive Gelfand--Tsetlin integral and the Dixon--Anderson recursion.  For Macdonald
  polynomials we prove the case \(n=2\), the covers \((d)\maj(d-1,1)\) for every \(n\) when \(t\leq q\),
  and, by exact certificates, the case \(n=3\) with \(\size\lambda\leq8\).  The
  HCIZ mechanism behind the Jack proof has no direct finite-\(q\) analogue:
  at two explicit pairs \((q,t)\), no positive representation of the normalized ratio has a kernel
  multiplicative in \(\lambda\), and normalizing at the principal specialization \(t^\delta\) instead
  fails the inequality.
\end{abstract}
\preprintslink{jackratio}{8 October 2026}
\maketitle

\section{Introduction}

A partition \(\lambda=(\lambda_1\geq\lambda_2\geq\cdots\geq0)\) has size
\(\size\lambda=\sum_i\lambda_i\); we pad partitions with zeros as needed.
For partitions with at most \(n\) parts we write \(\lambda\maj\mu\), and say
that \(\lambda\) \emph{majorizes} (or \emph{dominates}) \(\mu\), if
\begin{equation}
 \sum_{i=1}^r\lambda_i\geq\sum_{i=1}^r\mu_i\quad(1\leq r<n),
 \qquad
 \sum_{i=1}^n\lambda_i=\sum_{i=1}^n\mu_i.
 \label{eq:majorization}
\end{equation}
Thus \((3,1)\maj(2,2)\): the more ``spread out'' partition is the larger
one.  The same definition applies to arbitrary real vectors once they are
sorted in decreasing order; it is then the classical majorization order of
Hardy, Littlewood and P\'olya~\citep{marshall2011}.

For a partition \(\lambda\) with at most \(n\) parts, let
\(P_\lambda(x;q,t)\) be the monic Macdonald polynomial in the variables
\(x=(x_1,\ldots,x_n)\), as in \citet[Ch.~VI]{macdonald1995}, and let
\begin{equation}
 \F_\lambda^{(n)}(x;q,t)
 :=\frac{P_\lambda(x_1,\ldots,x_n;q,t)}{P_\lambda(1^n;q,t)},
 \qquad 0<q,t<1,\quad x\in\reals_{>0}^n,
 \label{eq:Fdef}
\end{equation}
be its normalization at the point \(1^n=(1,\ldots,1)\).  Unless a
degeneration is stated, \(0<q,t<1\); then \(P_\lambda(1^n;q,t)>0\) and
\(P_\lambda\) has nonnegative coefficients in the monomial basis
\citep[Ch.~VI, \S\S6--7]{macdonald1995}.  We study the conjecture
\begin{equation*}
 \lambda\maj\mu
 \quad\Longrightarrow\quad
 \F_\lambda^{(n)}(x;q,t)\geq \F_\mu^{(n)}(x;q,t)
 \qquad(x\in\reals_{>0}^n).
 \tag{M}
\end{equation*}
Both sides of (M) are homogeneous of the same degree and equal one at
\(x=1^n\), so (M) concerns the shape of \(\F_\lambda\) away from the
diagonal.  We call \eqref{eq:Fdef} the \emph{anchored} ratio, to
distinguish it from the principal-specialization ratio
\(P_\lambda(x)/P_\lambda(t^\delta)\), with \(t^\delta=(t^{n-1},\ldots,t,1)\),
which fails the analogue of (M) (Appendix~\ref{sec:omega}).

The Schur case \(q=t\) of (M) is the conjecture of \citet{cuttler2011},
proved by \citet[posted February 2015]{sra2016} with the
Harish-Chandra--Itzykson--Zuber (HCIZ) integral (Section~\ref{sec:hciz})
and, by a different method and for real exponents, by
\citet[Theorems~1.14 and~1.15]{khare2021}.  The same paper suggested
that the HCIZ approach should yield the analogous inequalities for zonal,
Jack and Macdonald polynomials; (M) is the Macdonald form of that
suggestion.  The Jack degeneration \(q\to1\), \(t=q^{1/\alpha}\), was later
stated as a conjecture by \citet[Conjecture~4.7]{mcswiggen2022} for
Heckman--Opdam polynomials of arbitrary root systems, with a proof in
semisimple rank one, and by \citet{chen2024}; it is
Theorem~\ref{thm:jack} below.  For Macdonald polynomials, \citet{chen2025}
stated (M) as a conjecture, together with several stronger positivity
statements (their Conjecture~5), and proved the cases \(n=2\) and
\(\mu=(1^{\size\lambda})\), \(n\geq\size\lambda\) (their Theorem~5.5).
\citet{mcswiggen2026log,mcswiggen2026} proved the Jack case
by two further routes and established finite-\(q\) results for the
principal-specialization normalization on shifted \(q\)-lattices;
Appendices~\ref{sec:omega} and~\ref{sec:audit} explain why those results do
not reach (M), and correct one of them.

\subsection{Results}\label{sec:results}
Our first theorem proves (M) for (monic) Jack polynomials \(P_\lambda(x;\alpha)\). 

\begin{theorem}[Jack majorization in every rank]\label{thm:jack}
For every \(\alpha>0\), \(n\geq1\), \(x\in\reals_{>0}^n\), and equal-size
partitions \(\lambda\maj\mu\) with at most \(n\) parts,
\[
 \frac{P_\lambda(x;\alpha)}{P_\lambda(1^n;\alpha)}
 \geq
 \frac{P_\mu(x;\alpha)}{P_\mu(1^n;\alpha)}.
\]
If \(\lambda\neq\mu\) and \(x_i\neq x_j\ \forall i\neq j\),
the inequality is strict; if \(x=c\one\), equality holds for all equal-size
\(\lambda,\mu\).
\end{theorem}

Its proof (Section~\ref{sec:jack}) uses only Sun's integral and the
Okounkov--Olshanski recursion, and differs from both proofs of
McSwiggen--Sahi (Remark~\ref{rmk:routes}).  At finite \(q\) we prove (M)
for \(n=2\) and all \(0<q,t<1\).

\begin{theorem}[Macdonald majorization in rank two]\label{thm:rank2}
For \(0<q,t<1\), \(x_1,x_2>0\), and equal-size two-part partitions
\(\lambda\maj\mu\),
\[
 \F_\lambda^{(2)}(x;q,t)\geq \F_\mu^{(2)}(x;q,t).
\]
If \(\lambda\ne\mu\) and \(x_1\ne x_2\), the inequality is strict.
\end{theorem}

The statement is Theorem~5.5 of \citet{chen2025}, who obtain the stronger
conclusion that \(\F_\lambda^{(2)}-\F_\mu^{(2)}\) lies in the Muirhead cone.
Our proof (Section~\ref{sec:finiteq}) uses a short induction on
Rogers' recurrence for the continuous \(q\)-ultraspherical polynomials.

For larger \(n\) we prove one family of covers and a finite range of
sizes.  For \(n\geq2\), \(d\geq2\), and \(0<t\leq q<1\),
\(\F_{(d)}\geq\F_{(1)}\F_{(d-1)}\), hence \(\F_{(d)}\geq\F_{(d-1,1)}\), and
the same holds for rectangular translates (Theorem~\ref{thm:tophook}).  The
proof is a likelihood-ratio argument in which the restriction \(t\leq q\)
is exactly the log-concavity of the coefficients of
\((tz;q)_\infty/(z;q)_\infty\).  For \(n=3\) we prove (M) for all
\(0<q,t<1\) and \(\size\lambda\leq8\) (Certificate~\ref{cert:allcovers}), by exact
tensor-Bernstein certificates over \(\mathbb Q(q,t)\) of every dominance
cover.  The three-row cover \((4,2)\maj(4,1,1)\) is certified strictly off
the diagonal (Certificate~\ref{cert:hard}).  The conjecture remains open
in general; Section~\ref{sec:status} lists cases beyond these results.

\subsection{The HCIZ mechanism, and what is missing at finite \texorpdfstring{\(q\)}{q}}\label{sec:hciz}

The proof of the Schur case in \citet{sra2016} and the proof of
Theorem~\ref{thm:jack} below rest on one mechanism.
Write \(x=e^{z}\) coordinatewise.  In the Schur case,
Harish-Chandra's formula for the unitary integral
\citep{harishchandra1957,itzykson1980} gives
\begin{equation}
 \frac{s_\lambda(e^{z})}{s_\lambda(1^n)}
 =\frac{\vdm(z)}{\vdm(e^{z})}
  \int_{U(n)}\exp\bigl(\trace(\diag(z)\,U\diag(\lambda+\delta)U^*)\bigr)\,dU,
 \qquad
 \delta=(n-1,n-2,\ldots,0),
 \label{eq:hciz}
\end{equation}
with \(dU\) the Haar probability measure and \(\vdm\) the Vandermonde
product; coincident coordinates are interpreted by continuity.
The integral is the Laplace transform, evaluated at
\(\lambda+\delta\), of the push-forward of Haar measure under
\(U\mapsto\diag(U^*\diag(z)U)\), a positive measure independent of
\(\lambda\).  This Laplace transform is log-convex and symmetric in its
argument, and a symmetric convex function is
Schur-convex \citep[Proposition~C.2]{marshall2011}.  Since
\(\lambda\maj\mu\) implies \(\lambda+\delta\maj\mu+\delta\), the Schur
case of (M) follows.  The same argument for spherical functions of
symmetric spaces is in \citet{mcswiggen2022}.

For Jack polynomials there is no unitary integral, but the mechanism
survives.  \citet{sun2014}, building on \citet{borodin2015}, wrote the
type-\(A\) Heckman--Opdam hypergeometric function \(\Fc_\theta(z,s)\), up to
a positive factor independent of \(s\), as the Laplace transform in \(s\)
of a positive measure on Gelfand--Tsetlin patterns, for every
\(\theta>0\); and \(\Fc_\theta(z,\lambda+\theta\rho)\), with \(\rho\) the
Weyl vector, is the normalized Jack polynomial.  Section~\ref{sec:jack}
proves Theorem~\ref{thm:jack} this way.  It obtains the specialization to
Jack polynomials by identifying the Gelfand--Tsetlin measure, tilted at
\(s=\theta\rho\), with the iterated Dixon--Anderson links of
\citet{okounkov1997}.

At finite \(q\) no positive integral of this kind is known.  Cherednik's
global spherical function is the bispectral eigenfunction of the Macdonald
difference operators, and \(\F_\lambda^{(n)}\) is its anchored ratio
(Section~\ref{sec:global}); what is missing is a positive representation
of that ratio in which \(\lambda\) enters through a kernel one can compare
along the majorization order.  One might replace \(e^{\ip{\lambda}{y}}\)
by a real kernel \(K_\lambda(y)\) with \(K_{\lambda+\mu}=K_\lambda K_\mu\).
Proposition~\ref{prop:hankel} and Corollary~\ref{cor:semichar} show that
this replacement fails: at two pairs \((q,t)\), one with \(t>q\) and one with
\(t<q\), the values \(\F_{(r)}^{(2)}(2,1;q,t)\) have a negative
\(3\times3\) Hankel determinant, so they are not the moments of any
positive measure on the line.  A kernel valid for all \(0<q,t<1\) must
therefore depend on \(\lambda\) nonmultiplicatively; finding one is the
``non-exponential HCIZ problem'' of the title, and
Section~\ref{sec:constraints} collects the constraints on it.  For
\(n\geq2\), no positive Doob transform makes the dual Macdonald operator a
positive combination of \(q\)-shifts on the positive chamber
(Proposition~\ref{prop:dual}).

The finite-\(q\) analogue of the Dixon--Anderson recursion that does
exist, Okounkov's \(q\)-integral formula \citep{okounkov1998}, is
normalized at \(t^\delta\) rather than at \(1^n\), and so are the lattice
results of \citet{mcswiggen2026log,mcswiggen2026}.  This normalization
fails: \(\lambda\mapsto P_\lambda(x)/P_\lambda(t^\delta)\) is neither
Schur-convex nor log-convex already at \(x=1^2\), a point of the dominant
\(q\)-lattice whenever \(t=q^k\) with \(k\in\Z_{\geq1}\)
(Proposition~\ref{prop:omega-fail}, Remark~\ref{rmk:omega-logconvex}).

A positive structure without multiplicativity does exist:
\(\lambda\mapsto\F_\lambda(X)\) is a positive harmonic function of an
Up--Down random walk on signatures (Proposition~\ref{prop:updown}).  For
\(n=3\), coupling the Up steps from \(\lambda\) and from
\(\lambda+e_1-e_2\) reduces to two explicit inequalities between
evaluations \(P_\nu(1^3;q,t)\) (Proposition~\ref{prop:hall}).  Total
positivity of the rank-three branching kernel is equivalent to an
inequality between the evaluations \(P_{(m)}(1,1;q,t)\) that is stronger
than their log-concavity when \(t<q\) (Remark~\ref{rmk:branching}).  These
inequalities are open; they hold on a grid of \((q,t)\) for small indices
(Remark~\ref{rmk:cuts-numeric}).

\subsection{Conventions}\label{sec:conventions}

For a vector \(v\) we write \(\size v=\sum_iv_i\) and
\(\vdm(v)=\prod_{i<j}(v_i-v_j)\).  The \(q\)-shifted factorial is
\((a;q)_k=\prod_{j=0}^{k-1}(1-aq^j)\), with \((a;q)_\infty\) the infinite
product.  We write \(e_n(x)=x_1\cdots x_n\)
and \(p_1(x)=x_1+\cdots+x_n\).  The rectangular-translation identity
\(P_{\lambda+(r^n)}=e_n^rP_\lambda\) \citep[Ch.~VI, \S4]{macdonald1995}
and \(e_n(1^n)=1\) give \(\F_{\lambda+(r^n)}=e_n^r\F_\lambda\).  Vectors
\(z\in\reals^n\) are \emph{regular} if their coordinates are pairwise
distinct, and \emph{central} if they are all equal.  Jack polynomials
\(P_\lambda(x;\alpha)\) are monic, with \(\theta=1/\alpha\), and
\(\F_\lambda^{(n)}(x;\alpha)=P_\lambda(x;\alpha)/P_\lambda(1^n;\alpha)\) is
the Jack analogue of \eqref{eq:Fdef}; such ratios do not depend on the
choice among the usual scalar normalizations.

Statements labeled \emph{exact certificate} concern finitely many explicit
rational functions of \((q,t)\); their proofs are finite computations in
exact arithmetic over \(\mathbb Q(q,t)\), done in SageMath.

\section{The Jack case}\label{sec:jack}

Throughout this section \(\theta>0\) and \(\alpha=1/\theta\).  The proof of
Theorem~\ref{thm:jack} has two steps.  First, for fixed \(z\), the function
\(s\mapsto\Fc_\theta(z,s)\) is symmetric and is a positive multiple of the
Laplace transform of a positive measure independent of \(s\) (Sun's
integral, Subsection~\ref{sec:sun}).  Second, \(\Fc_\theta(z,\lambda+\theta\rho)\)
is the normalized Jack polynomial (Theorem~\ref{thm:DA}).  A symmetric
convex function is Schur-convex, and \(\lambda\maj\mu\) implies
\(\lambda+\theta\rho\maj\mu+\theta\rho\), so the two steps give the
inequality; Subsection~\ref{sec:jackthm} also proves the strictness
clause.

\subsection{Sun's Gelfand--Tsetlin integral}\label{sec:sun}

Let
\[
 \rho=\left(\frac{n-1}{2},\frac{n-3}{2},\ldots,\frac{1-n}{2}\right),
 \qquad \size\rho=0.
\]
For \(z_1>\cdots>z_n\), a Gelfand--Tsetlin pattern below \(z\) is an array
\[
 \xi^{(1)}\intl\xi^{(2)}\intl\cdots\intl\xi^{(n)}=z,
 \qquad
 \xi_i^{(\ell+1)}\geq\xi_i^{(\ell)}\geq\xi_{i+1}^{(\ell+1)},
\]
where \(\xi^{(\ell)}\in\reals^\ell\); the set of all such patterns is the
Gelfand--Tsetlin polytope \(\GT(z)\), of dimension \(n(n-1)/2\).  Put
\(\xi^{(0)}=\varnothing\) and
\begin{equation}
 R_\ell(\xi)=\size{\xi^{(\ell)}}-\size{\xi^{(\ell-1)}}
 \qquad(1\leq\ell\leq n),
 \label{eq:Rdef}
\end{equation}
so that \(R(\xi)=(R_1(\xi),\ldots,R_n(\xi))\) is a linear image of the
pattern with \(\size{R(\xi)}=\size z\).  Following
\citet[(1.6) and Theorem~1.4]{sun2014}, define
\begin{align}
 \Phi_\theta(z,s)
 &:=\Gamma(\theta)^{-n(n-1)/2}
   \int_{\GT(z)}e^{\ip{s}{R(\xi)}}\,
   W^0_{\theta,z}(\xi)\,d\xi,                                      \label{eq:phi}\\
 W^0_{\theta,z}(\xi)
 &:=\prod_{\ell=1}^{n-1}
 \frac{
  \displaystyle\prod_{i=1}^{\ell}\prod_{j=1}^{\ell+1}
  \left|e^{\xi_i^{(\ell)}}-e^{\xi_j^{(\ell+1)}}\right|^{\theta-1}}
 {\displaystyle
  \prod_{1\leq i<j\leq\ell}
  \left|e^{\xi_i^{(\ell)}}-e^{\xi_j^{(\ell)}}\right|^{\theta-1}
  \prod_{1\leq i<j\leq\ell+1}
  \left|e^{\xi_i^{(\ell+1)}}-e^{\xi_j^{(\ell+1)}}\right|^{\theta-1}}
 e^{-(\theta-1)\size{\xi^{(\ell)}}}.                             \label{eq:weight}
\end{align}
All factors in \eqref{eq:weight} are nonnegative, and the Selberg-type
singularities are locally integrable for every \(\theta>0\).  Sun's
Theorem~1.4, which Sun attributes to the Macdonald limit of
\citet[Definition~6.1 and Proposition~6.3]{borodin2015}, states that
\begin{equation}
 \Fc_\theta(z,s)
 =\frac{\prod_{j=1}^n\Gamma(j\theta)}
 {\Gamma(\theta)^n
  \prod_{i<j}\left(2\sinh\frac{z_i-z_j}{2}\right)^\theta}
 \Phi_\theta(z,s)                                                  \label{eq:ho}
\end{equation}
is the type-\(A\) Heckman--Opdam hypergeometric function
\citep{heckman1987,heckman1994}, in the normalization
\(\Fc_\theta(0,s)=1\).  It is real-analytic, symmetric in the coordinates
of each argument, and a joint eigenfunction of the Heckman--Opdam
operators; the second-order one is
\begin{equation}
 \mathcal D_\theta
 =\sum_{i=1}^n\partial_{z_i}^2
 +\theta\sum_{i<j}\coth\!\left(\frac{z_i-z_j}{2}\right)
       (\partial_{z_i}-\partial_{z_j}),
 \qquad
 \mathcal D_\theta\Fc_\theta(z,s)
 =\bigl(\norm{s}^2-\theta^2\norm{\rho}^2\bigr)\Fc_\theta(z,s).      \label{eq:deigen}
\end{equation}
The proof of Theorem~\ref{thm:jack} uses three features of
\eqref{eq:phi}--\eqref{eq:ho}: \(W^0_{\theta,z}\geq0\); the spectral
variable \(s\) enters only through \(e^{\ip{s}{R(\xi)}}\); and
\(\Fc_\theta(z,s)\) is symmetric in \(s\).

\begin{rmk}[The central \(\mathrm{GL}_n\) coordinate]\label{rmk:central}
There is no trace-zero restriction in \eqref{eq:ho}.  Translating every row
of a pattern by \(c\) maps \(\GT(z)\) onto \(\GT(z+c\one)\), adds \(c\) to
every \(R_\ell\), and leaves \(W^0\) and the prefactor in \eqref{eq:ho}
unchanged (the exponents of \(e^{c}\) in \eqref{eq:weight} cancel level by
level); and \(\size{R(\xi)}=\size z\).  Hence
\[
 \Fc_\theta(z+c\one,s)=e^{c\size s}\Fc_\theta(z,s),\qquad
 \Fc_\theta(z,s+c\one)=e^{c\size z}\Fc_\theta(z,s),
\]
and, writing \(\bar z=z-(\size z/n)\one\) for the trace-zero part,
\begin{equation}
 \Fc_\theta(z,s)=e^{\size z\,\size s/n}\,\Fc_\theta(\bar z,\bar s).
                                                                   \label{eq:central}
\end{equation}
At \(s=\lambda+\theta\rho\), identity \eqref{eq:central} is the homogeneity
of Jack polynomials.
\end{rmk}

\subsection{Dixon--Anderson links and the spectral specialization}\label{sec:DA}

For \(x_1>\cdots>x_n>0\), write \(y\intl x\) if \(y\in\reals^{n-1}\) and
\(x_1\geq y_1\geq x_2\geq\cdots\geq y_{n-1}\geq x_n\), and define the
\emph{Dixon--Anderson link}
\begin{equation}
 \Lambda_{n-1,n}^\theta(x,dy)
 :=\frac{\Gamma(n\theta)}{\Gamma(\theta)^n}
   \frac{\vdm(y)}{\vdm(x)^{2\theta-1}}
   \prod_{i=1}^{n-1}\prod_{j=1}^n
   \abs{y_i-x_j}^{\theta-1}\,dy
 \qquad(y\intl x).                                                 \label{eq:dalink}
\end{equation}
The Dixon--Anderson integral \citep{anderson1991} says that
\eqref{eq:dalink} has total mass one for every \(\theta>0\), so
\(\Lambda_{n-1,n}^\theta(x,\cdot)\) is a probability measure on
\(\{y\intl x\}\).  Jack polynomials are its eigenfunctions in the following
sense.

\begin{theorem}[\citealt{okounkov1997}, \S6]\label{thm:OO}
Let \(\nu\) be a partition with at most \(n-1\) parts.  Then
\begin{equation}
 \int_{y\intl x}P_\nu(y;1/\theta)\,\Lambda_{n-1,n}^\theta(x,dy)
 =\frac{C_\nu^{(n)}}{C_0^{(n)}}P_\nu(x;1/\theta),
 \qquad
 C_\nu^{(n)}=\prod_{i=1}^{n-1}
 B\bigl(\nu_i+(n-i)\theta,\theta\bigr),                              \label{eq:daeig}
\end{equation}
where \(B\) is the beta function and
\(C_0^{(n)}=\Gamma(\theta)^n/\Gamma(n\theta)\).
\end{theorem}

Okounkov and Olshanski obtain \eqref{eq:daeig} as the \(q\to1\) limit of
Okounkov's \(q\)-integral formula \citep{okounkov1998}; \(C_0^{(n)}\) is
the Dixon--Anderson integral itself.  Letting \(x\to1^n\) in \eqref{eq:daeig}, the link concentrates at
\(y=1^{n-1}\), so \(C_\nu^{(n)}/C_0^{(n)}=P_\nu(1^{n-1})/P_\nu(1^n)\) and
\eqref{eq:daeig} becomes
\begin{equation}
 \frac{P_\nu(x;1/\theta)}{P_\nu(1^n;1/\theta)}
 =\int_{y\intl x}
   \frac{P_\nu(y;1/\theta)}{P_\nu(1^{n-1};1/\theta)}
   \Lambda_{n-1,n}^\theta(x,dy).                                   \label{eq:darec}
\end{equation}
Identity \eqref{eq:darec} is the \(\theta\)-analogue of the branching
rule.  Its finite-\(q\) analogue, with the anchored normalization, is the
object that Section~\ref{sec:global} shows to be missing
(Definition~\ref{def:link}).

Iterating \eqref{eq:darec} produces a random Gelfand--Tsetlin pattern.  Let
\(X^{(n)}=x\), and for \(k=n,n-1,\ldots,2\) let \(X^{(k-1)}\) be drawn from
\(\Lambda_{k-1,k}^\theta(X^{(k)},\cdot)\); write \(\E_x^\theta\) for the
resulting expectation, and put
\begin{equation}
 D_k=\prod_{i=1}^kX_i^{(k)},\qquad D_0=1,\qquad
 R_k=\log\frac{D_k}{D_{k-1}}\quad(1\leq k\leq n),                  \label{eq:DR}
\end{equation}
so that \(R=(R_1,\ldots,R_n)\) is the vector \eqref{eq:Rdef} of the pattern
\(\xi_i^{(k)}=\log X_i^{(k)}\).

\begin{theorem}[Jack ratios as Dixon--Anderson averages]\label{thm:DA}
Let \(x_1>\cdots>x_n>0\) and \(z=\log x\).  For every partition \(\lambda\)
with at most \(n\) parts, with \(\lambda_{n+1}:=0\),
\begin{equation}
 \frac{P_\lambda(x;1/\theta)}{P_\lambda(1^n;1/\theta)}
 =\E_x^\theta\prod_{k=1}^n D_k^{\lambda_k-\lambda_{k+1}}
 =\E_x^\theta\,e^{\ip{\lambda}{R}}.                                 \label{eq:genpow}
\end{equation}
Moreover, the law of the pattern \(\xi=\log X\) under \(\E_x^\theta\) is the
Gelfand--Tsetlin measure of \eqref{eq:phi} tilted at \(s=\theta\rho\):
\begin{equation}
 \E_x^\theta\,g(\xi)
 =\frac{\prod_{j=1}^n\Gamma(j\theta)}
       {\Gamma(\theta)^{n(n+1)/2}\prod_{i<j}\bigl(2\sinh\frac{z_i-z_j}{2}\bigr)^\theta}
  \int_{\GT(z)}g(\xi)\,e^{\ip{\theta\rho}{R(\xi)}}\,W^0_{\theta,z}(\xi)\,d\xi
 \label{eq:tilt}
\end{equation}
for every bounded measurable \(g\).  Consequently
\begin{equation}
 \frac{P_\lambda(e^z;1/\theta)}{P_\lambda(1^n;1/\theta)}
 =\Fc_\theta(z,\lambda+\theta\rho).                                 \label{eq:spec}
\end{equation}
\end{theorem}

\begin{proof}
Let \(\lambda'=\lambda-(\lambda_n^n)\), a partition with at most \(n-1\)
parts.  Since \(P_\lambda=e_n^{\lambda_n}P_{\lambda'}\) and
\(e_n(1^n)=1\), we have
\(\F_\lambda^{(n)}(x;1/\theta)=D_n^{\lambda_n}\F_{\lambda'}^{(n)}(x;1/\theta)\),
and \eqref{eq:darec} gives
\(\F_{\lambda'}^{(n)}(x;1/\theta)=\E_x^\theta\,\F_{\lambda'}^{(n-1)}(X^{(n-1)};1/\theta)\).
Iterating down the pattern gives the first equality in \eqref{eq:genpow};
the second is the definition of \(R\).

For \eqref{eq:tilt}, change variables \(X_i^{(k)}=e^{\xi_i^{(k)}}\) in the
product of the links.  The constants multiply to
\(\prod_{k=2}^n\Gamma(k\theta)/\Gamma(\theta)^k
 =\prod_{j=1}^n\Gamma(j\theta)/\Gamma(\theta)^{n(n+1)/2}\), which is the
constant in \eqref{eq:tilt}.  The Vandermonde factors multiply to
\(\prod_{k=1}^{n-1}\vdm(X^{(k)})^{2-2\theta}\cdot\vdm(x)^{1-2\theta}\),
because \(\vdm(X^{(1)})=1\) and each intermediate level appears once as
a ``\(y\)'' and once as an ``\(x\)''.  The corresponding product in
\eqref{eq:weight}, after \(X=e^\xi\), is
\(\prod_{k=1}^{n-1}\vdm(X^{(k)})^{2-2\theta}\cdot\vdm(x)^{1-\theta}\),
and the cross factors \(\prod\abs{X_i^{(k)}-X_j^{(k+1)}}^{\theta-1}\)
agree.  The Jacobian is \(\prod_{k<n}\prod_idX_i^{(k)}
=e^{\sum_{k<n}\size{\xi^{(k)}}}d\xi\), while \eqref{eq:weight} carries
\(e^{-(\theta-1)\sum_{k<n}\size{\xi^{(k)}}}\).  Altogether,
\[
 \prod_{k=2}^n\Lambda_{k-1,k}^\theta(X^{(k)},dX^{(k-1)})
 =\frac{\prod_{j=1}^n\Gamma(j\theta)}{\Gamma(\theta)^{n(n+1)/2}}
  \,\vdm(x)^{-\theta}\,
  e^{\theta\sum_{k=1}^{n-1}\size{\xi^{(k)}}}\,W^0_{\theta,z}(\xi)\,d\xi.
\]
Since \(\rho_k-\rho_{k+1}=1\) for \(k<n\) and \(\rho_n=(1-n)/2\), Abel
summation gives
\(\ip{\theta\rho}{R(\xi)}=\theta\sum_{k<n}\size{\xi^{(k)}}-\theta\frac{n-1}{2}\size z\),
and
\(\prod_{i<j}2\sinh\frac{z_i-z_j}{2}=\vdm(x)\,e^{-\frac{n-1}{2}\size z}\).
Substituting both, the product of the links equals the right-hand side of
\eqref{eq:tilt} with \(g\equiv1\), which proves \eqref{eq:tilt}.  Finally,
taking \(g=e^{\ip{\lambda}{R}}\) in \eqref{eq:tilt} and comparing with
\eqref{eq:phi}--\eqref{eq:ho} gives \(\E_x^\theta e^{\ip{\lambda}{R}}
=\Fc_\theta(z,\lambda+\theta\rho)\), which with \eqref{eq:genpow} is
\eqref{eq:spec}.
\end{proof}

Identity \eqref{eq:spec} is classical: the Heckman--Opdam function at the
spectral point \(\lambda+\theta\rho\) is the normalized Jacobi polynomial,
which in type \(A\) is the Jack polynomial \citep[(4.4.10)]{heckman1994};
see also \citet[(4.16)]{mcswiggen2022}.  The proof above is included
because it shows how the two positive kernels fit together.  The
Gelfand--Tsetlin integral \eqref{eq:phi} is symmetric in \(s\), though its
weight need not have mass one; the Dixon--Anderson links are probability
measures but not symmetric in \(\lambda\); and the shift by \(\theta\rho\)
converts one into the other.

\subsection{The majorization theorem}\label{sec:jackthm}

\begin{proof}[\textbf{Proof of Theorem~\ref{thm:jack}}]
\emph{\textbf{The inequality.}}  First suppose that \(x\) is regular.
By symmetry we may sort \(x\); put \(z=\log x\).
By \eqref{eq:phi}--\eqref{eq:ho}, for fixed \(z\) the function
\(s\mapsto\Fc_\theta(z,s)\) is \(c_\theta(z)\int e^{\ip{s}{R(\xi)}}d\omega(\xi)\)
for a positive constant \(c_\theta(z)\) and a positive measure
\(\omega=W^0_{\theta,z}\,d\xi\) on \(\GT(z)\), independent of \(s\).  A
Laplace transform of a positive measure is log-convex, hence convex; and
\(\Fc_\theta(z,\cdot)\) is symmetric.  A symmetric convex function is
Schur-convex \citep[Proposition~C.2]{marshall2011}.  The vectors
\(\lambda+\theta\rho\) and \(\mu+\theta\rho\) are sorted, and adding
\(\theta\rho\) to both sides of each inequality in \eqref{eq:majorization}
preserves it, so \(\lambda\maj\mu\) implies
\(\lambda+\theta\rho\maj\mu+\theta\rho\).  Hence
\(\Fc_\theta(z,\lambda+\theta\rho)\geq\Fc_\theta(z,\mu+\theta\rho)\), which
by \eqref{eq:spec} is the inequality.  For non-regular \(x\) the inequality
follows by continuity of both sides in \(x\).

\emph{\textbf{Equality.}}  If \(x=c\one\), then
\(P_\nu(x;\alpha)=c^{\size\nu}P_\nu(1^n;\alpha)\) by homogeneity, so both
ratios equal \(c^{\size\nu}\), and \(\size\lambda=\size\mu\).

\emph{\textbf{Strictness.}}  Let \(x\) be regular and \(n\geq2\), so that \(z\) is
regular and not central.  Differentiating under the integral,
\[
 \nabla_s^2\Fc_\theta(z,s)
 =c_\theta(z)\int_{\GT(z)}R(\xi)R(\xi)^{\mathsf T}e^{\ip{s}{R(\xi)}}\,
  d\omega(\xi),
\]
so for \(v\in\reals^n\),
\(v^{\mathsf T}\nabla_s^2\Fc_\theta(z,s)\,v
 =c_\theta(z)\int\ip{v}{R(\xi)}^2e^{\ip{s}{R(\xi)}}d\omega(\xi)\geq0\), with
equality only if \(\ip{v}{R(\xi)}=0\) for \(\omega\)-almost every \(\xi\).
Since \(\omega\) has a density that is positive almost everywhere on the
full-dimensional polytope \(\GT(z)\), and \(\xi\mapsto\ip{v}{R(\xi)}\) is
affine, equality forces \(\ip{v}{R(\xi)}=0\) on all of \(\GT(z)\).  Now the
vertices of \(\GT(z)\) at which each row is obtained from the row above by
deleting one entry have \(R(\xi)\) equal to a permutation of \(z\), and every
permutation of \(z\) arises this way.  For non-central \(z\) the
permutations of \(z\) affinely span the hyperplane
\(\{\size w=\size z\}\), whose direction space is \(\one^\perp\).  Hence
\(\ip{v}{R}\equiv0\) on \(\GT(z)\) forces \(v\in\reals\one\).  So the
Hessian of \(\Fc_\theta(z,\cdot)\) is positive definite on \(\one^\perp\) at
every \(s\), and \(\Fc_\theta(z,\cdot)\) is strictly convex on every affine
hyperplane \(\{\size s=\text{const}\}\).

A strictly convex symmetric function on such a hyperplane is strictly
Schur-convex there: if \(a\maj b\) with \(b\) not a permutation of \(a\),
then \(b\) is a convex combination \(\sum_kp_k\sigma_k(a)\) of permutations of
\(a\) \citep[Chapter~2, B.2]{marshall2011}, at least two of which are distinct
and carry positive weight, so strict convexity gives
\(f(b)<\sum_kp_kf(\sigma_ka)=f(a)\).  Apply strict Schur-convexity with
\(a=\lambda+\theta\rho\) and \(b=\mu+\theta\rho\): both are sorted, so they
are permutations of each other only if \(\lambda=\mu\).
\end{proof}

\begin{rmk}[Three routes to the Jack inequality]\label{rmk:routes}
The inequality of Theorem~\ref{thm:jack}, without strictness, is
Corollary~2.4 of \citet{mcswiggen2026log}, follows from Theorem~2.8 there,
and follows from Theorem~4.8 of \citet{mcswiggen2026}.  Theorem~2.8 proves
the Schur-convexity of \(s\mapsto\Fc_\theta(z,s)\) by induction on \(n\)
from the positive interlacing recursion of Amri and Bedhiafi for the
nonsymmetric Opdam function; Theorem~4.8 deduces it from a positive Laplace
representation of the Opdam--Cherednik kernel for all root systems.  The
proof above is a third route, and it gives the strictness clause directly.
Corollary~2.4 of \citet{mcswiggen2026log} rests on a finite-\(q\) lattice
theorem that fails at central points (Appendix~\ref{sec:audit}), but
their Theorem~2.8 proves the Jack statement independently.
\end{rmk}

\begin{rmk}[Why a pointwise comparison fails]\label{rmk:robinhood}
Formula \eqref{eq:genpow} invites the attempt to prove (M) in the Jack case
pointwise: under a root transfer \(\lambda\mapsto\lambda+e_i-e_j\), \(i<j\),
the integrand \(\prod_kD_k^{\lambda_k-\lambda_{k+1}}\) is multiplied by
\(D_iD_{j-1}/(D_{i-1}D_j)\), and one might hope that
\(D_iD_{j-1}\geq D_{i-1}D_j\).  This inequality fails already for \(n=2\) and
\((2,0)\maj(1,1)\): the two integrands are \(Y^2\) and \(x_1x_2\), where the
Dixon--Anderson (beta) variable \(Y=X^{(1)}_1\) has positive mass on both
sides of \(\sqrt{x_1x_2}\).  The proof of Theorem~\ref{thm:jack} uses
convexity and symmetry of the full average instead; item~(\ref{item:averages})
of Section~\ref{sec:constraints} records this constraint.
\end{rmk}

\section{Rank two}\label{sec:finiteq}\label{sec:rank2}

The proof of Theorem~\ref{thm:rank2} is an induction on Rogers' three-term
recurrence for the continuous \(q\)-ultraspherical polynomials, which are
the one-row Macdonald polynomials in two variables.  The same recurrence
drives the results of Appendix~\ref{app:finiteq}.

Put
\begin{equation}
 w_k=\frac{(t;q)_k}{(q;q)_k}\qquad(k\geq0),
 \label{eq:wk}
\end{equation}
the coefficients of \((tz;q)_\infty/(z;q)_\infty=\sum_kw_kz^k\).  The
one-row Macdonald polynomial in two variables is
\(P_{(m)}(x_1,x_2;q,t)=w_m^{-1}\sum_{j=0}^mw_jw_{m-j}x_1^{m-j}x_2^j\)
\citep[Ch.~VI, \S\S2 and 4]{macdonald1995}.  Writing \(x_1=re^{u}\),
\(x_2=re^{-u}\), the polynomial \(P_{(m)}\) is \(r^m\) times the continuous
\(q\)-ultraspherical polynomial
\begin{equation}
 C_m(\cos\vartheta;t\mid q)
 =\sum_{j=0}^m w_jw_{m-j}\,e^{\mathrm i(m-2j)\vartheta}
 \label{eq:qultra}
\end{equation}
\citep[\S14.10.1]{koekoek2010hypergeometric}, evaluated at
\(\cos\vartheta=\cosh u\), divided by \(w_m\).  Let \(c_m=C_m(1;t\mid q)
=\sum_jw_jw_{m-j}>0\).

\begin{proof}[\textbf{Proof of Theorem~\ref{thm:rank2}}]
For \(\lambda=(a,b)\), set \(d=a+b\), \(m=a-b\), and write
\(x_1=re^u\), \(x_2=re^{-u}\) with \(r=\sqrt{x_1x_2}\), so that
\(z:=\cosh u\geq1\), with \(z=1\) exactly when \(x_1=x_2\).  Since
\(P_{(a,b)}=(x_1x_2)^bP_{(m)}\),
\begin{equation}
 \F_{(a,b)}^{(2)}(x;q,t)=r^d\phi_m(z),
 \qquad \phi_m(z):=\frac{C_m(z;t\mid q)}{c_m}.                       \label{eq:r2reduce}
\end{equation}
(Here \(C_m(z)\) with \(z=\cosh u\) means the Laurent polynomial
\eqref{eq:qultra} with \(e^{\mathrm i\vartheta}\) replaced by \(e^u\); since
\(d\) and \(m\) have the same parity, every term \(r^de^{(m-2j)u}\) is a
monomial in \(x_1,x_2\).)  The cover \((a,b)\maj(a-1,b+1)\) keeps \(d\)
and lowers \(m\) by two, so it suffices to prove \(\phi_{m+1}\geq\phi_m\)
for \(z\geq1\), strictly for \(z>1\).

Rogers' recurrence \citep[\S14.10.1]{koekoek2010hypergeometric} reads
\begin{equation}
 2(1-tq^m)\,z\,C_m=(1-q^{m+1})C_{m+1}+(1-t^2q^{m-1})C_{m-1}
 \qquad(m\geq1).                                                    \label{eq:rogers}
\end{equation}
Divide by \(2(1-tq^m)c_m\) and put
\begin{equation}
 A_m=\frac{(1-q^{m+1})c_{m+1}}{2(1-tq^m)c_m},
 \qquad
 B_m=\frac{(1-t^2q^{m-1})c_{m-1}}{2(1-tq^m)c_m},
 \qquad A_0=1,\ B_0=0.                                             \label{eq:abcoef}
\end{equation}
For \(m\geq1\), \(A_m,B_m>0\), and evaluating \eqref{eq:rogers} at \(z=1\)
gives \(A_m+B_m=1\).  Hence \(z\phi_m=A_m\phi_{m+1}+B_m\phi_{m-1}\), that is,
\begin{equation}
 A_m(\phi_{m+1}-\phi_m)=(z-1)\phi_m+B_m(\phi_m-\phi_{m-1}).           \label{eq:phirec}
\end{equation}
Now \(\phi_m(z)>0\) for \(z\geq1\), because \eqref{eq:qultra} has positive
coefficients, and \(\phi_0=1\), \(\phi_1=z\).  Induction on \(m\) in
\eqref{eq:phirec} gives \(\phi_{m+1}-\phi_m\geq0\), with strict inequality
for \(z>1\) since then \((z-1)\phi_m>0\).
\end{proof}

\section{Exact rank-three certificates}\label{sec:certificates}

For \(n=3\), Theorem~\ref{thm:tophook} proves only the covers
\((d)\maj(d-1,1)\) and their rectangular translates, and only for
\(t\leq q\).  This section proves (M) for \(n=3\), all \(0<q,t<1\), and
\(\size\lambda\leq8\), and the cover \((4,2)\maj(4,1,1)\) strictly off the
diagonal, by exact computation over \(\mathbb Q(q,t)\).

By symmetry and homogeneity, it suffices to consider
\begin{equation}
 x=\bigl((1+u)(1+v),\ 1+v,\ 1\bigr),\qquad u,v\geq0,                \label{eq:chamber}
\end{equation}
which parametrizes the sorted chamber \(x_1\geq x_2\geq x_3=1\), with
\(u=v=0\) exactly at the diagonal.  For a pair \(\lambda\maj\mu\) of
partitions with at most three parts, the \emph{cleared gap}
\begin{equation}
 \widetilde G_{\lambda,\mu}(u,v)
 :=P_\lambda(x)\,P_\mu(1^3)-P_\mu(x)\,P_\lambda(1^3),
 \qquad
 \widetilde G_{\lambda,\mu}
 =P_\lambda(1^3)P_\mu(1^3)\bigl(\F_\lambda-\F_\mu\bigr),               \label{eq:cleared}
\end{equation}
is a polynomial in \(u,v\) whose coefficients, the \emph{chamber
coefficients}, are rational functions of \((q,t)\).  Since
\(P_\lambda(1^3)P_\mu(1^3)>0\), the inequality \(\F_\lambda\geq\F_\mu\)
holds if and only if \(\widetilde G_{\lambda,\mu}\geq0\) for all
\(u,v\geq0\), and it holds if every chamber coefficient is nonnegative on
\((0,1)^2\).

We write each chamber coefficient in lowest terms as \(N/D\) with
\(N,D\in\mathbb Q[q,t]\).  Every denominator \(D\) below factors into
monomials and polynomials \(h(q^at^b)\) with \(h\) having no root in
\((0,1)\) (in practice \(1-qt\), \(1-q^2t\), \(1-q^3t^2\) and \(1+qt\)), so
\(D\) has constant sign on the open square.  Orient \(N\) so that \(D>0\)
there, and expand \(N\) in the tensor-Bernstein basis of its bidegree
\((r,s)\),
\[
 N(q,t)=\sum_{a=0}^r\sum_{b=0}^s
 \beta_{ab}\binom raq^a(1-q)^{r-a}
               \binom sbt^b(1-t)^{s-b}.
\]
If every \(\beta_{ab}\geq0\), then \(N\geq0\) on \([0,1]^2\); since every
basis function is positive on \((0,1)^2\), a nonzero such \(N\) is positive
there.  The test is sufficient but not necessary, so it cannot certify a
false inequality; no degree elevation was needed below.  The counts of
Bernstein coefficients reported below include those of the oriented
denominators, which are expanded in the same way.

\subsection{The cover \texorpdfstring{\((4,2)\maj(4,1,1)\)}{(4,2) > (4,1,1)}}\label{sec:hardcover}

\begin{certificate}[The cover \((4,2)\maj(4,1,1)\)]\label{cert:hard}
For every \(0<q,t<1\) and \(x\in\reals_{>0}^3\),
\begin{equation}
 \F_{(4,2)}^{(3)}(x;q,t)
 \geq \F_{(4,1,1)}^{(3)}(x;q,t),                                     \label{eq:hardcover}
\end{equation}
with equality only when \(x_1=x_2=x_3\).
\end{certificate}

\begin{proof}[Exact certificate]
The cleared gap \eqref{eq:cleared} has \(32\) nonzero chamber
coefficients; the coefficients of \(u\) and \(v\) vanish.  Every
denominator factors as described above, and the Bernstein coefficients of
the numerators and oriented denominators are nonnegative: \(2966\)
positive and \(620\) zero.  Hence every chamber coefficient is nonnegative
on \((0,1)^2\), which proves \eqref{eq:hardcover}.  The coefficients of
\(u^2\) and \(v^2\) are nonzero, hence positive on \((0,1)^2\); since all
terms are nonnegative, the gap vanishes only at \(u=v=0\).
\end{proof}

\subsection{All covers of size at most eight}\label{sec:allcovers}

The method of Certificate~\ref{cert:hard} applies to every cover,
including the three-row covers \((a,b,0)\maj(a,b-1,1)\), which are not
rectangular translates of two-row covers.  Since dominance is the
transitive closure of its covers, certifying every cover of a given size
proves (M) for all pairs of that size.

\begin{certificate}[Every rank-three cover of size at most eight]\label{cert:allcovers}
For every dominance cover \(\lambda\maj\mu\) with \(\size\lambda\leq8\)
and at most three parts, every chamber coefficient of the cleared gap
\eqref{eq:cleared} is nonnegative on \((0,1)^2\).  Consequently (M) holds
for \(n=3\), all \(0<q,t<1\), and \(\size\lambda\leq8\).
\end{certificate}

\begin{proof}[Exact certificate]
There are \(36\) such covers, with between \(6\) and \(78\) nonzero
chamber coefficients each.  Each cleared gap is certified as in
Certificate~\ref{cert:hard}; the Bernstein test decided every coefficient
without degree elevation.
\end{proof}

\section{The global spherical function and the missing kernel}\label{sec:global}

The proof of Theorem~\ref{thm:jack} has two inputs: an eigenfunction whose
anchored ratio is \(\F_\lambda\), and a positive integral representation of
that ratio in which \(\lambda\) enters through an exponential.  At finite
\(q\) the first input exists: it is Cherednik's global spherical function
(Subsection~\ref{sec:anchor}).  This section shows that the second input
cannot take the forms suggested by the Jack case.  An anchored analogue of
\eqref{eq:darec}, and more generally a kernel multiplicative in
\(\lambda\), fails at two explicit pairs \((q,t)\)
(Subsections~\ref{sec:links}--\ref{sec:hankel}); a positive operator in the
spectral variable fails for every \(n\geq2\) (Subsection~\ref{sec:dual}).

\subsection{Cherednik's global spherical function}\label{sec:anchor}

In \(n\) variables, the first Macdonald--Ruijsenaars operator is
\begin{equation}
 D_1^{(n)}
 =\sum_{i=1}^n\prod_{j\neq i}
 \frac{tX_i-X_j}{X_i-X_j}\,T_{q,X_i},
 \qquad
 D_1^{(n)}P_\lambda
 =\Bigl(\sum_{i=1}^nq^{\lambda_i}t^{n-i}\Bigr)P_\lambda,           \label{eq:mrop}
\end{equation}
where \(T_{q,X_i}\) replaces \(X_i\) by \(qX_i\).  Cherednik's global
symmetric spherical function \(\Pc^\circ(X,\Lambda)\) is the
Weyl-invariant, bispectral reproducing kernel of the full commuting family
of the corresponding type-\(A\) operators \citep{cherednik2009}; its
Gaussian normalization removes the periodic ambiguity that the difference
equations alone leave.  In Cherednik's anti-dominant convention, let
\(c_\lambda=w_0(\lambda-(\size\lambda/n)\one)\), where \(w_0\) reverses
coordinates, and put \(\Lambda_\lambda=q^{c_\lambda-\rho_k}\), with
\(\rho_k\) the deformed Weyl vector of \citet{cherednik2009}.  Equation
(3.11) of \citet{cherednik2009}, on the type-\(A\) torus \(e_n(\bar X)=1\),
has the form
\begin{equation}
 \Pc^\circ(\bar X,\Lambda_\lambda)
 =C(q,t)\frac{P_\lambda(\bar X;q,t)}
                  {P_\lambda(q^{-\rho_k};q,t)},                    \label{eq:cherednik}
\end{equation}
where \(C(q,t)\) is independent of \(X\) and \(\lambda\).  Dividing
\eqref{eq:cherednik} by its value at \(\bar X=1^n\) removes every
convention-dependent constant.  To recover the \(\mathrm{GL}_n\) ratio,
put \(r=e_n(X)^{1/n}\) and \(\bar X=r^{-1}X\).

\begin{prop}[Anchored global spherical ratio]\label{prop:anchor}
For \(n\geq2\), every partition \(\lambda\) with at most \(n\) parts, and
\(X\in\reals_{>0}^n\),
\begin{equation}
 \F_\lambda^{(n)}(X;q,t)
 =r^{\size\lambda}\frac{\Pc^\circ(\bar X,\Lambda_\lambda)}
        {\Pc^\circ(1^n,\Lambda_\lambda)}.                          \label{eq:anchor}
\end{equation}
\end{prop}

\begin{proof}
Divide \eqref{eq:cherednik} by its value at \(\bar X=1^n\) and use
\(P_\lambda(X)=r^{\size\lambda}P_\lambda(\bar X)\).
\end{proof}

For equal-size partitions the factor \(r^{\size\lambda}\) is common to both
sides of (M), as in \eqref{eq:central}.  What is missing is an analogue of
the positive formula \eqref{eq:ho}.

\subsection{Anchored links}\label{sec:links}

The normalization of the Dixon--Anderson recursion \eqref{eq:darec}
suggests the following finite-\(q\) analogue.

\begin{defn}[Anchored link]\label{def:link}
A family \(\mathsf B_{n\to n-1}(dY\mid X)\) of positive probability measures
on \(\reals_{>0}^{n-1}\), indexed by \(X\in\reals_{>0}^n\), is an
\emph{anchored link} if for every partition \(\nu\) with at most \(n-1\)
parts
\begin{equation}
 \F_\nu^{(n)}(X;q,t)
 =\int \F_\nu^{(n-1)}(Y;q,t)\,\mathsf B_{n\to n-1}(dY\mid X).       \label{eq:linkf}
\end{equation}
\end{defn}

Multiplying \eqref{eq:linkf} by \(P_\nu(1^n)\) shows that an anchored link
is a positive kernel with the Macdonald polynomials as eigenfunctions and
with prescribed eigenvalues,
\begin{equation}
 \int P_\nu(Y;q,t)\,\mathsf B_{n\to n-1}(dY\mid X)
 =\frac{P_\nu(1^{n-1};q,t)}{P_\nu(1^n;q,t)}\,P_\nu(X;q,t),           \label{eq:link}
\end{equation}
and conversely.  Since the \(P_\nu\) span the symmetric polynomials,
\eqref{eq:link} determines the link on polynomials; the question is whether
the functional it defines is positive.  For \(t=q^K\), Okounkov's
\(q\)-integral formula \citep[Theorem~I]{okounkov1998} satisfies an
identity of the form \eqref{eq:link}, but with the principal ratios
\(P_\nu(t^{\delta_{n-1}})/P_\nu(t^{\delta_n})\) as eigenvalues, where
\(t^{\delta_k}=(t^{k-1},\ldots,t,1)\): at \(x=(1,t,\ldots,t^{n-1})\) only one
summand of the \(q\)-integral survives \citep[Example~3.11]{okounkov1998}.
Iterating an anchored link down to one variable would give
\(\F_\lambda^{(n)}(X)=\E\prod_kD_k^{\lambda_k-\lambda_{k+1}}\) exactly as
in \eqref{eq:genpow}, a positive representation of \(\F_\lambda\) with a
kernel multiplicative in \(\lambda\).  The next subsection rules out such
a representation at two pairs \((q,t)\).

\subsection{The Hankel obstruction}\label{sec:hankel}

\begin{prop}[Hankel obstruction]\label{prop:hankel}
Let \(m_r=\F_{(r)}^{(2)}(2,1;q,t)\), \(r\geq0\).  At \((q,t)=(\tfrac13,\tfrac78)\)
and at \((q,t)=(\tfrac13,\tfrac1{11})\) there is no positive measure on
\(\reals\) whose moments are \(m_0,\ldots,m_4\).
\end{prop}

\begin{proof}
For every positive measure \(\nu\) on \(\reals\) with moments \(m_r\), the
Hankel matrix \(H=(m_{i+j})_{i,j=0}^2\) is positive semidefinite, because
\(c^{\mathsf T}Hc=\int(c_0+c_1y+c_2y^2)^2\,d\nu(y)\geq0\).  Exact
computation from the two-variable polynomials \(P_{(r)}\) gives, at
\(q=\tfrac13\), \(t=\tfrac78\),
\begin{equation}
 (m_0,\ldots,m_4)
 =\Bigl(1,\tfrac32,\tfrac{93}{38},\tfrac{17}{4},\tfrac{4089}{532}\Bigr),
 \qquad
 \det H=-\tfrac{67}{192052}<0,                                        \label{eq:hankela}
\end{equation}
and at \(q=\tfrac13\), \(t=\tfrac1{11}\),
\begin{equation}
 (m_0,\ldots,m_4)
 =\Bigl(1,\tfrac32,\tfrac{30}{13},\tfrac{277}{76},\tfrac{65721}{11126}\Bigr),
 \qquad
 \det H=-\tfrac{832225}{70593757936}<0.                               \label{eq:hankelb}
\end{equation}
Explicit witnesses are \(c=(13,-20,7)\), with \(c^{\mathsf T}Hc=-\tfrac1{76}\)
in the first case, and \(c=(19,-25,8)\), with
\(c^{\mathsf T}Hc=-\tfrac{3030}{1374061}\) in the second.
\end{proof}

The two pairs have \(t>q\) and \(t<q\), so the obstruction is not confined
to either side of the Schur line \(t=q\).  In the Jack limit, by contrast,
the values \(\F_{(r)}^{(2)}(2,1)\) are the moments of a Dixon--Anderson
beta law on \([1,2]\).

\begin{corr}[No anchored link in rank two; no multiplicative kernel]\label{cor:semichar}
At the two parameter pairs of Proposition~\ref{prop:hankel}:
\begin{enumerate}
\item there is no anchored link \(\mathsf B_{2\to1}\) in the sense of
Definition~\ref{def:link};
\item more generally, there is no representation
\(\F_\lambda^{(2)}(X)=\int K_\lambda(y)\,\nu_X(dy)\) valid for all
partitions \(\lambda\), with \(\nu_X\) a positive measure and a real kernel
that is multiplicative in the partition, \(K_{\lambda+\mu}=K_\lambda K_\mu\).
\end{enumerate}
\end{corr}

\begin{proof}
Since \(\F_{(r)}^{(1)}(Y)=Y^r\), an anchored link \(\mathsf B_{2\to1}\)
would make \(m_r=\int Y^r\,\mathsf B_{2\to1}(dY\mid(2,1))\) the moments of
a positive measure.  For the second statement, multiplicativity gives
\(K_0^2=K_0\) and \(K_0K_\lambda=K_\lambda\).  Restrict \(\nu_X\) to
\(\{K_0=1\}\); all kernels vanish on its complement, and the restricted
measure has mass \(\F_0(X)=1\).  Along the ray \(\lambda=re_1\), the values
\(\F_{re_1}^{(2)}(X)=\int K_{e_1}(y)^r\nu_X(dy)\) are the moments of the
push-forward of \(\nu_X\) under \(K_{e_1}\), a positive measure on
\(\reals\).  Both contradict Proposition~\ref{prop:hankel} at \(X=(2,1)\).
\end{proof}

The path representation of Appendix~\ref{sec:updown} is not multiplicative
in \(\lambda\), so Corollary~\ref{cor:semichar} does not apply to it.

\subsection{The dual difference equation is not termwise positive}\label{sec:dual}

Cherednik's function is bispectral, so one may equally try to generate
\(\lambda\mapsto \F_\lambda(X)\) by a positive operator acting in the
spectral variable.  The type-\(A\) dual Macdonald operator is
\begin{equation}
 D_\Lambda=\sum_{i=1}^n A_i(\Lambda)T_{q,\Lambda_i},
 \qquad
 A_i(\Lambda)=\prod_{j\ne i}
 \frac{t\Lambda_i-\Lambda_j}{\Lambda_i-\Lambda_j}.                 \label{eq:dualop}
\end{equation}

\begin{prop}[No positive Doob transform of the dual operator]\label{prop:dual}
Let \(n\geq2\), \(0<q,t<1\), and let \(h\) be a positive function on
\(\reals_{>0}^n\).  Then the coefficients \(a_i\) of
\(h^{-1}D_\Lambda h=\sum_ia_i(\Lambda)T_{q,\Lambda_i}\) are not all
nonnegative on the chamber \(\Lambda_1>\cdots>\Lambda_n>0\).
\end{prop}

\begin{proof}
The transformed coefficient is
\(a_i(\Lambda)=A_i(\Lambda)\,h(T_{q,\Lambda_i}\Lambda)/h(\Lambda)\), which
has the sign of \(A_i(\Lambda)\).  Choose
\(t\Lambda_1<\Lambda_2<\Lambda_1\), and, when \(n>2\), choose
\(t\Lambda_1>\Lambda_3>\cdots>\Lambda_n>0\).
The factor indexed by \(j=2\) in \(A_1\) is negative and every other factor
is positive, so \(A_1(\Lambda)<0\).
\end{proof}

Positivity in the spectral variable can therefore emerge only after
restriction to a lattice that avoids the region
\(t\Lambda_1<\Lambda_2<\Lambda_1\), or after composing several steps; it
cannot be read off term by term from the difference equation.

\section{Constraints on a \texorpdfstring{\((q,t)\)}{(q,t)}-HCIZ kernel}\label{sec:constraints}

A finite-\(q\) construction combining the HCIZ comparison with the
rank-reducing recursion of Section~\ref{sec:jack} would need to:
\begin{enumerate}
\item have the eigenvalues \(P_\nu(1^{n-1})/P_\nu(1^n)\) of
\eqref{eq:link}, not the principal ratios
\(P_\nu(t^{\delta_{n-1}})/P_\nu(t^{\delta_n})\) of Okounkov's
\(q\)-integral formula; the ratio normalized at \(t^\delta\) fails the
inequality (Appendix~\ref{sec:omega});
\item\label{item:averages} compare averages rather than integrands, since
\(D_iD_{j-1}\geq D_{i-1}D_j\) fails already in the Jack case
(Remark~\ref{rmk:robinhood});
\item depend on \(\lambda\) nonmultiplicatively, whatever the support of
the measure (Corollary~\ref{cor:semichar});
\item\label{item:limits} reduce to the HCIZ integral \eqref{eq:hciz} at \(q=t\), to the
Dixon--Anderson links \eqref{eq:dalink} as \(q\to1\), and for \(n=2\) to
the recurrence \eqref{eq:rogers}.
\end{enumerate}
The Up--Down walk of Proposition~\ref{prop:updown} gives a positive path
representation of \(\F_\lambda(X)\) for every non-central
\(X\in\reals_{>0}^n\), and its Martin representation \eqref{eq:martin} is
not multiplicative in \(\lambda\).  Its boundary and its limits in
item~(\ref{item:limits}) have not been identified, and it does not supply the
rank-reducing link \eqref{eq:link}, which Corollary~\ref{cor:semichar}
rules out at the pairs of Proposition~\ref{prop:hankel}.  A proof of (M)
based on the walk would therefore need a comparison argument on
signatures.

\section{Open problems}\label{sec:status}

\begin{enumerate}
\item Conjecture (M) beyond the cases \(q=t\); \(\mu=(1^{\size\lambda})\)
with \(n\geq\size\lambda\); \(n=2\); the covers \((d)\maj(d-1,1)\) with
\(t\leq q\); and \(n=3\) with \(\size\lambda\leq8\).  Open cases include
\((8,1)\maj(7,2)\) for \(n=3\) and all \(0<q,t<1\), and
\((3,1)\maj(2,2)\) for \(n=4\) and \(q\neq t\).
\item Inequality \eqref{eq:strongbarrier}, equivalent to the
\(\mathrm{MTP}_2\) property of the rank-three branching kernel, and the
inequalities \eqref{eq:Lcut}--\eqref{eq:Rcut} for the Up coupling of
\(\lambda\) and \(\lambda+e_1-e_2\) when \(n=3\)
(Proposition~\ref{prop:hall}); also whether \eqref{eq:strongbarrier}
implies \eqref{eq:Lcut}--\eqref{eq:Rcut}.  All three inequalities hold in
the tests of Remark~\ref{rmk:cuts-numeric}.
\item The Martin boundary of the Up--Down walk (Remark~\ref{rmk:martin}),
which in the formulation of this paper is the finite-\(q\) HCIZ problem.
\item Whether the product inequality \eqref{eq:tophook} holds for \(t>q\)
when \(n\geq3\); no counterexample was found.
\end{enumerate}

\appendix

\section{Further results at finite \texorpdfstring{\(q\)}{q}}\label{app:finiteq}

This appendix proves the remaining finite-\(q\) results: log-concavity of
the evaluations \(P_{(m)}(1,1;q,t)\) and the inequality
\(\F_{(d)}\geq\F_{(d-1,1)}\) for \(t\leq q\).  Both rest on Rogers'
recurrence \eqref{eq:rogers}.  The main text does not use them.

\subsection{Log-concavity of the one-row evaluations}\label{sec:logconcave}

Let
\begin{equation}
 \ev{m}:=P_{(m)}(1,1;q,t)=\frac{c_m}{w_m}\qquad(m\geq0),
 \label{eq:Deltam}
\end{equation}
with \(\ev{0}=1\) and \(\ev{1}=2\).  For \(n=2\), the Up step of the walk
in Appendix~\ref{app:updown} adds a box to the first row of \((m,0)\) with
probability \(\ev{m+1}/(2\ev{m})\); the following proposition shows that
these probabilities are non-increasing in \(m\).

\begin{prop}[Log-concavity of \(\ev{m}\)]\label{prop:logconcave}
For every \(0<q,t<1\), the sequence \((\ev{m})_{m\geq0}\) is log-concave:
\(\ev{m}^2\geq\ev{m-1}\ev{m+1}\) for \(m\geq1\).  Equivalently,
\(r_m:=\ev{m}/\ev{m-1}\) is non-increasing in \(m\).
\end{prop}

\begin{proof}
Rogers' recurrence \eqref{eq:rogers} at \(z=1\), divided by \(w_m\) and
using \(w_{m\pm1}/w_m\) from \eqref{eq:wk}, becomes a recurrence for
\(\ev{m}\):
\[
 \ev{m}=2\ev{m-1}-\kappa_m\ev{m-2},\qquad
 \kappa_m=\frac{(1-t^2q^{m-2})(1-q^{m-1})}
 {(1-tq^{m-1})(1-tq^{m-2})}\quad(m\geq2).
\]
Put \(s_m=r_m-1\) and
\[
 \varepsilon_m=1-\kappa_m
 =\frac{q^{m-2}(1-t)(q-t)}
 {(1-tq^{m-1})(1-tq^{m-2})}.
\]
Then \(s_1=1\), and dividing the recurrence by \(\ev{m-1}\) gives
\[
 s_m=\frac{s_{m-1}+\varepsilon_m}{1+s_{m-1}},\qquad
 s_{m-1}-s_m=\frac{s_{m-1}^2-\varepsilon_m}{1+s_{m-1}}.
\]
If \(t\geq q\), then \(\varepsilon_m\leq0\) and \(s_{m-1}-s_m\geq0\)
follows at once.  Suppose \(t<q\).  Then \(0<\varepsilon_m<1\), and
\(\varepsilon_{m+1}/\varepsilon_m=q(1-tq^{m-2})/(1-tq^m)<1\), so
\((\varepsilon_m)\) is decreasing.  We claim \(s_{m-1}\geq\sqrt{\varepsilon_m}\)
for all \(m\geq2\), which gives \(s_{m-1}-s_m\geq0\).  For \(m=2\) the claim reads
\(1\geq\sqrt{\varepsilon_2}\).  For the induction step, the recurrence gives
\[
 s_m-\sqrt{\varepsilon_m}
 =\frac{(s_{m-1}-\sqrt{\varepsilon_m})(1-\sqrt{\varepsilon_m})}{1+s_{m-1}}
 \geq0,
\]
so \(s_m\geq\sqrt{\varepsilon_m}\geq\sqrt{\varepsilon_{m+1}}\).  In all
cases \(r_m\leq r_{m-1}\).
\end{proof}

\subsection{The covers \texorpdfstring{\((d)\maj(d-1,1)\)}{(d) > (d-1,1)} for \texorpdfstring{\(t\leq q\)}{t <= q}}\label{sec:tophook}

Theorem~\ref{thm:tophook} below compares consecutive one-row polynomials.
Up to scalars, the one-row polynomials are the coefficients of a product of
generating functions, so total positivity of convolutions applies.  The
polynomials \(Q_{(m)}=w_mP_{(m)}\) have the generating function
\begin{equation}
 \sum_{m\geq0}Q_{(m)}(x)z^m
 =\prod_{i=1}^nW(x_iz),
 \qquad
 W(z)=\frac{(tz;q)_\infty}{(z;q)_\infty}=\sum_{k\geq0}w_kz^k,
 \label{eq:genfun}
\end{equation}
\citep[Ch.~VI, \S2]{macdonald1995}, so \(Q_{(m)}(x)\) is the
\(m\)-th coefficient of the convolution of the \(n\) sequences
\((w_kx_i^k)_{k\geq0}\).  Two classical facts about such convolutions are
needed.  A nonnegative sequence with no internal zeros is called
log-concave (or \(\mathrm{PF}_2\)) if \(h_k^2\geq h_{k-1}h_{k+1}\); a
sequence \((f'_k)\) dominates \((f_k)\) in the \emph{likelihood-ratio
order}, written \(f'\geq_{\mathrm{lr}}f\), if \(f'_k/f_k\) is non-decreasing
in \(k\).  Convolution with a \(\mathrm{PF}_2\) sequence preserves both
log-concavity and the likelihood-ratio order
\citep[Chapter~3]{karlin1968}; see \citet[Theorem~1.C.9]{shaked2007} for the
second statement.  Both are instances of the basic composition formula for
totally positive kernels.

\begin{theorem}[The covers \((d)\maj(d-1,1)\) for \(t\leq q\)]\label{thm:tophook}
Let \(n\geq2\), \(d\geq2\), \(x\in\reals_{>0}^n\), and \(0<t\leq q<1\).  Then
\begin{equation}
 \F_{(d)}^{(n)}(x;q,t)
 \geq \F_{(1)}^{(n)}(x)\,\F_{(d-1)}^{(n)}(x;q,t)                         \label{eq:tophook}
\end{equation}
and consequently
\begin{equation}
 \F_{(d)}^{(n)}(x;q,t)\geq \F_{(d-1,1)}^{(n)}(x;q,t).                  \label{eq:tophookcover}
\end{equation}
Both inequalities are strict unless \(x_1=\cdots=x_n\).  For every integer
\(r\geq0\), rectangular translation gives
\begin{equation}
 \F_{(d+r,r^{n-1})}^{(n)}
 \geq \F_{(d-1+r,1+r,r^{n-2})}^{(n)},                                \label{eq:tophookrect}
\end{equation}
again strictly away from the diagonal.
\end{theorem}

\begin{proof}
\emph{\textbf{Step 1: two variables.}}  For \(a,b>0\) put \(e_1=a+b\), \(e_2=ab\),
and \(f_m=f_m(e_2):=[z^m]\,W(az)W(bz)\), a polynomial in \(e_1,e_2\).  With
\(a=re^u\), \(b=re^{-u}\) we get \(f_m=r^mC_m(\cosh u)\), so Rogers' recurrence
\eqref{eq:rogers} at index \(m-1\), multiplied by \(r^m\), gives, for
\(m\geq1\) and \(f_{-1}=0\),
\begin{equation}
 (1-q^m)f_m
 =(1-tq^{m-1})\,e_1f_{m-1}
 -(1-t^2q^{m-2})\,e_2f_{m-2}.                                         \label{eq:frec}
\end{equation}
Fix \(e_1\) and regard \(e_2\in(0,e_1^2/4]\) as the variable; a
two-coordinate spread \((a,b)\mapsto(a+\epsilon,b-\epsilon)\) with
\(a\geq b\) keeps \(e_1\) and lowers \(e_2\).  Let \(u_m=f_m/f_{m-1}\),
\(\gamma_m=(1-tq^{m-1})/(1-q^m)\) and \(\eta_m=(1-t^2q^{m-2})/(1-q^m)\).
Then \(u_1=\gamma_1e_1\) does not depend on \(e_2\), and for \(m\geq2\),
\eqref{eq:frec} gives
\[
 u_m=\gamma_me_1-\frac{\eta_me_2}{u_{m-1}},\qquad
 \frac{du_m}{de_2}=-\frac{\eta_m\bigl(u_{m-1}-e_2\,u_{m-1}'\bigr)}{u_{m-1}^2}.
\]
Since \(\eta_m>0\), \(u_{m-1}>0\) and, inductively, \(u_{m-1}'\leq0\), we get
\(u_m'<0\) for all \(m\geq2\).  Hence if \(e_2'<e_2\) then
\(u_m(e_2')\geq u_m(e_2)\), strictly for \(m\geq2\); so the ratio
\(f_m(e_2')/f_m(e_2)=\prod_{k\leq m}u_k(e_2')/u_k(e_2)\) is non-decreasing
in \(m\), and strictly increasing for \(m\geq2\).  Thus a spread of two
coordinates raises the coefficient sequence of \(W(az)W(bz)\) in the
likelihood-ratio order.

\emph{\textbf{Step 2: the other variables.}}  The coefficient sequence
\((w_kx_i^k)_k\) of each remaining factor \(W(x_iz)\) is log-concave
exactly when
\begin{equation}
 \frac{w_{k+1}}{w_k}=\frac{1-tq^k}{1-q^{k+1}}                       \label{eq:wratio}
\end{equation}
is non-increasing in \(k\), and a brief calculation shows that
\(w_{k+1}/w_k\geq w_{k+2}/w_{k+1}\) is equivalent to
\((q-t)(1-q)q^k\geq0\).  So for \(t\leq q\) every factor is
\(\mathrm{PF}_2\), and so is the convolution \(h\) of all factors other
than the two being spread.  Convolving the likelihood-ratio comparison of
Step~1 with \(h\) preserves it, so a two-coordinate spread raises the
sequence \((Q_{(m)}(x))_{m\geq0}\) in the likelihood-ratio order.
Strictness survives the convolution: by the basic composition formula, the
\(2\times2\) minor expressing the comparison after convolution is a
nonnegative combination of the minors before it, and the term indexed by
the two top positions carries the weight \(h_0^2>0\).

\emph{\textbf{Step 3: chaining.}}  Let \(\bar x=n^{-1}\sum_ix_i\).  Since
\(x/\bar x\maj1^n\), the vector \(x/\bar x\) is obtained from
\(1^n\) by finitely many two-coordinate spreads, each keeping the
coordinates positive \citep[Chapter~2, B.1]{marshall2011}.  Along the chain
the sequence \((Q_{(m)})_m\) increases in the likelihood-ratio order, hence
\(Q_{(d)}(x/\bar x)/Q_{(d)}(1^n)\geq Q_{(d-1)}(x/\bar x)/Q_{(d-1)}(1^n)\),
which is \(\F_{(d)}(x/\bar x)\geq \F_{(d-1)}(x/\bar x)\).  Homogeneity and
\(\F_{(1)}(x)=\bar x\) turn this inequality into \eqref{eq:tophook}, with strict
inequality as soon as one genuine spread occurs, that is, unless
\(x=\bar x\one\).

\emph{\textbf{Step 4: the cover.}}  The Pieri rule for multiplication by
\(e_1=p_1\) \citep[Ch.~VI, (6.24)]{macdonald1995} has positive coefficients
for \(0<q,t<1\), and evaluating it at \(1^n\) normalizes them:
\begin{equation}
 \F_{(1)}\F_{(d-1)}
 =\sigma_{d,n}\F_{(d)}+(1-\sigma_{d,n})\F_{(d-1,1)},
 \qquad 0<\sigma_{d,n}<1.                                            \label{eq:pieri}
\end{equation}
Substituting \eqref{eq:pieri} into \eqref{eq:tophook} and cancelling
\(1-\sigma_{d,n}\) gives \eqref{eq:tophookcover}.  Finally
\(P_{\lambda+(r^n)}=e_n^rP_\lambda\) with \(e_n(1^n)=1\) gives
\eqref{eq:tophookrect}.
\end{proof}

The restriction \(t\leq q\) enters only in Step~2; for \(n=2\) there are
no other variables, and the argument gives \eqref{eq:tophook} for all
\(0<q,t<1\).  Whether \eqref{eq:tophook} holds for \(t>q\) when \(n\geq3\)
is open; exact checks for \(n=3\), \(d\leq5\), a few points \(x\), and all
\((q,t)\in\{i/12:1\leq i\leq11\}^2\) found no violation.

\section{The Up--Down walk and the rank-three coupling}\label{app:updown}

Corollary~\ref{cor:semichar} rules out positive representations of
\(\F_\lambda\) that are multiplicative in \(\lambda\).  This appendix
constructs one that is not: a random walk on signatures with
\(\lambda\mapsto\F_\lambda(X)\) as a positive harmonic function.  It then
reduces one family of couplings of this walk for \(n=3\) to two open
inequalities between evaluations \(P_\nu(1^3;q,t)\).

\subsection{A positive Up--Down walk with \texorpdfstring{\(\F_\lambda\)}{F} as harmonic function}\label{sec:updown}

Fix \(n\) and let \(\mathcal S\) be the set of dominant integer signatures
\(\lambda\in\Z^n\), \(\lambda_1\geq\cdots\geq\lambda_n\).  Extend Macdonald
polynomials to signatures by
\(P_\lambda:=e_n^{-r}P_{\lambda+(r^n)}\) for any \(r\) with
\(\lambda+(r^n)\) a partition; the rectangular-translation identity makes
the definition independent of \(r\).  Put \(\ev{\lambda}=P_\lambda(1^n;q,t)>0\), so
that \(\F_\lambda=P_\lambda/\ev{\lambda}\) as before.  For \(X\in\reals_{>0}^n\) set
\begin{equation}
 \Xi(X)=\frac{p_1(X)\,p_1(X^{-1})}{n^2}
 =\frac{(\sum_iX_i)(\sum_iX_i^{-1})}{n^2}\ \geq\ 1,                  \label{eq:Xi}
\end{equation}
with equality exactly when \(X\) is central, by the Cauchy--Schwarz
inequality.

\begin{prop}[The Up--Down walk]\label{prop:updown}
Let \(0<q,t<1\).
\begin{enumerate}
\item (Up step.)  Writing \(p_1P_\lambda=\sum_{\nu=\lambda+\square}c_{\lambda\nu}P_\nu\)
for the Pieri rule, the matrix
\(\mathsf U_{\lambda\nu}=c_{\lambda\nu}\ev{\nu}/(n\ev{\lambda})\) is stochastic on
\(\mathcal S\), and \(\sum_\nu\mathsf U_{\lambda\nu}\F_\nu(X)=(p_1(X)/n)\,\F_\lambda(X)\).
\item (Down step.)  Writing
\(p_1(X^{-1})P_\lambda(X)=\sum_{\nu=\lambda-\square}c'_{\lambda\nu}P_\nu(X)\),
the matrix \(\mathsf D_{\lambda\nu}=c'_{\lambda\nu}\ev{\nu}/(n\ev{\lambda})\) is
stochastic, and \(\sum_\nu\mathsf D_{\lambda\nu}\F_\nu(X)=(p_1(X^{-1})/n)\,\F_\lambda(X)\).
\item (Harmonicity.)  Let \(\mathsf K=\mathsf D\mathsf U\), a stochastic
kernel on each component \(\mathcal S_c=\{\lambda:\size\lambda=c\}\).  Then
\begin{equation}
 \sum_\nu\mathsf K(\lambda,\nu)\F_\nu(X)=\Xi(X)\,\F_\lambda(X),         \label{eq:updowneig}
\end{equation}
so that for non-central \(X\), \(\lambda\mapsto \F_\lambda(X)\) is a positive
harmonic function of the strictly substochastic kernel
\(\mathsf P_X=\mathsf K/\Xi(X)\).
\item (Feynman--Kac.)  Let \((\lambda(r))_{r\geq0}\) be the Markov chain
with kernel \(\mathsf K\) and \(\lambda(0)=\lambda\), let \(D\subset\mathcal S_c\)
be finite with \(\lambda\in D\), and let \(\tau_D\) be the first exit time
from \(D\).  If \(X\) is not central, then
\begin{equation}
 \F_\lambda(X)
 =\E_\lambda\!\left[
   \Xi(X)^{-\tau_D}\F_{\lambda(\tau_D)}(X)\,;\ \tau_D<\infty\right].   \label{eq:feynkac}
\end{equation}
\end{enumerate}
\end{prop}

\begin{proof}
(1) The Pieri coefficients \(c_{\lambda\nu}\) are nonnegative for
\(0<q,t<1\) \citep[Ch.~VI, (6.24)]{macdonald1995}, and they are unchanged
by rectangular translation, so the expansion holds for signatures.
Evaluating it at \(1^n\) gives
\(n\ev{\lambda}=\sum_\nu c_{\lambda\nu}\ev{\nu}\), which is the row-sum
condition; dividing the expansion by \(n\ev{\lambda}\) gives the
eigenrelation.

(2) The inversion identity
\(P_\lambda(X^{-1};q,t)=e_n(X)^{-\lambda_1}P_{\lambda_1\one-w_0\lambda}(X;q,t)\),
with \(w_0\lambda=(\lambda_n,\ldots,\lambda_1)\), holds for Macdonald
polynomials \citep[Ch.~VI, \S4]{macdonald1995}; in the signature notation
it reads \(P_\lambda(X^{-1})=P_{\lambda^*}(X)\) with
\(\lambda^*=-w_0\lambda\).  Applying (1) to \(\lambda^*\) and substituting
\(X\mapsto X^{-1}\) gives the Down expansion with
\(c'_{\lambda\nu}=c_{\lambda^*\nu^*}\geq0\), and
\(\nu^*=\lambda^*+\square\) is the same as \(\nu=\lambda-\square\).  The
rest is as in (1).

(3) Compose the two eigenrelations.  Each step changes \(\size\lambda\) by
\(\pm1\), so \(\mathsf K\) preserves \(\mathcal S_c\).

(4) By \eqref{eq:updowneig}, \(M_r=\Xi(X)^{-r}\F_{\lambda(r)}(X)\) is a
martingale for the \(\mathsf K\)-chain, so
\(\F_\lambda(X)=\E_\lambda M_{\tau_D\wedge r}\) for every \(r\).  On
\(\{\tau_D\leq r\}\), \(M_{\tau_D\wedge r}=\Xi^{-\tau_D}\F_{\lambda(\tau_D)}\)
is bounded by the maximum of \(\F\) over the finitely many states reachable
in one step from \(D\); on \(\{\tau_D>r\}\), \(\lambda(r)\in D\) and
\(M_r\leq\Xi^{-r}\max_D\F\to0\) because \(\Xi(X)>1\).  Dominated convergence
as \(r\to\infty\) gives \eqref{eq:feynkac}.
\end{proof}

\begin{rmk}[The Martin boundary as the finite-\(q\) HCIZ integral]\label{rmk:martin}
Fix non-central \(X\), so that \(\mathsf P_X\) is transient, since
\(\sum_{r\geq0}\mathsf P_X^r(\lambda,\nu)\leq\sum_{r\geq0}\Xi(X)^{-r}<\infty\).
Whenever \(\mathsf P_X\) is irreducible on \(\mathcal S_c\), the minimal
Martin boundary theory of transient chains \citep[Chapter~IV]{woess2000}
represents the positive harmonic function of
Proposition~\ref{prop:updown}(3) as
\begin{equation}
 \F_\lambda^{(n)}(X;q,t)
 =\int_{\partial_M^{\min}}\mathsf K_X^{\mathrm{Martin}}(\lambda,\zeta)
       \,\nu_X(d\zeta)                                              \label{eq:martin}
\end{equation}
for a positive measure \(\nu_X\) on the minimal Martin boundary of
\(\mathsf P_X\).  This representation is on a fixed-size component and
imposes no multiplicativity in \(\lambda\), so it does not contradict
Corollary~\ref{cor:semichar}, which concerns a single multiplicative
representation for all partitions.
Identifying the boundary, the kernel and the measure explicitly is, in
this formulation, the finite-\(q\) HCIZ problem, and \eqref{eq:martin} by
itself says nothing about majorization.  The walk does offer a coupling
strategy, which we now describe.
\end{rmk}

\subsection{An adjacent-row Up coupling reduces to two evaluation inequalities}\label{sec:hall}

Call \((\lambda,\mu)\) a \emph{root pair} if \(\mu=\lambda+e_i-e_j\) for
some \(i<j\) with both signatures dominant; then \(\mu\maj\lambda\), and
every pair \(\mu\maj\lambda\) is connected by a chain of root pairs.  Suppose
that for every root pair the Up steps from \(\lambda\) and from \(\mu\)
can be coupled so that the outcomes are equal or again a root pair, and
likewise for the Down steps.  Then the gaps
\(G(\lambda,\mu)=\F_\mu-\F_\lambda\) satisfy
\(\Xi\,G(\lambda,\mu)=\sum\tilde{\mathsf K}((\lambda,\mu),(\lambda',\mu'))G(\lambda',\mu')\),
where \(\tilde{\mathsf K}\) is the coupled kernel and the sum runs over the
coupled outcomes; equal outcomes contribute zero.  This positive recursion
is the first step toward a maximum-principle argument for (M), and for
each root pair the existence of the coupling is a finite question.  We
treat the Up coupling for \(n=3\) and \(\mu=\lambda+e_1-e_2\).  The other
root transfers, and the boundary control needed for a maximum principle,
remain open.

For integers \(A\geq0\) and \(B\geq1\) put
\[
 \lambda=(A+B,B,0),\qquad
 \mu=(A+B+1,B-1,0),\qquad
 \ev{A,B}=P_{(A+B,B)}(1^3;q,t),
\]
with \(\ev{A,0}=P_{(A)}(1^3)\) (so \(\ev{A,B}=\ev{(A+B,B)}\) in the
notation of Appendix~\ref{sec:updown}), and let
\begin{equation}
 \varpi(A,B)=\psi_2(A+B)\,\psi_1(B),\qquad
 \psi_h(g)=
 \frac{(1-q^{g-1}t^{h+1})(1-q^gt^{h-1})}
      {(1-q^gt^h)(1-q^{g-1}t^h)},                                    \label{eq:varpi}
\end{equation}
which is the Pieri coefficient of \(P_{(A+B,B,1)}\) in
\(p_1P_{(A+B,B,0)}\).

\begin{prop}[Hall reduction for \(n=3\)]\label{prop:hall}
For the root pair \((\lambda,\mu)\) above, a coupling of the one-box Up
steps from \(\lambda\) and from \(\mu\) that sends every pair of outcomes to
equal signatures or to a root pair exists if and only if
\begin{align}
 L_{A,B}
 &:=\ev{A+1,B}\ev{A+2,B-1}
    -\ev{A,B}\ev{A+3,B-1}\geq0,                              \label{eq:Lcut}\\
 R_{A,B}
 &:=\varpi(A,B)\,\ev{A,B-1}\ev{A+2,B-1}
   -\varpi(A+2,B-1)\,\ev{A+2,B-2}\ev{A,B}\geq0,             \label{eq:Rcut}
\end{align}
where for \(B=1\) the second term of \(R_{A,1}\) is defined to be zero.
The same conditions characterize the Down coupling for the dual root pair
\((\lambda^*,\mu^*)=(-w_0\lambda,-w_0\mu)\).
\end{prop}

\begin{proof}
Label the outcome of an Up step by the row \(i\in\{1,2,3\}\) that receives
the box, and let \(p_i\) and \(p_i'\) be the probabilities of outcome \(i\)
from \(\lambda\) and from \(\mu\).  The pairs of outcomes that are equal or
root pairs are
\[
 1\longmapsto\{1,2\},\qquad
 2\longmapsto\{2\},\qquad
 3\longmapsto\{2,3\}
\]
(rows of \(\lambda\) on the left, admissible rows of \(\mu\) on the right;
\(\lambda+e_1=\mu+e_2\) is the equal pair), after deleting any outcome
that has probability zero.  A coupling supported on these pairs is a
transport plan between \(p\) and \(p'\) with the given bipartite support,
and by the supply--demand form of Hall's theorem
\citep{strassen1965} it exists if and only if \(p(S)\leq p'(N(S))\) for
every set \(S\) of rows, \(N(S)\) being its neighborhood.  The conditions
for \(S=\{2,3\}\) and \(S=\{1,2\}\) read \(p_1\geq p'_1\) and
\(p_3\geq p'_3\); given these two, the remaining conditions follow from
\(p_1+p_2+p_3=p'_1+p'_2+p'_3=1\).

The monic top Pieri coefficient and rectangular translation give
\[
 p_1=\frac{\ev{A+1,B}}{3\ev{A,B}},\qquad
 p'_1=\frac{\ev{A+3,B-1}}{3\ev{A+2,B-1}},
\]
while the bottom Pieri coefficient \eqref{eq:varpi} and translation by
\(-(1^3)\) give
\[
 p_3=\frac{\varpi(A,B)\ev{A,B-1}}{3\ev{A,B}},\qquad
 p'_3=\frac{\varpi(A+2,B-1)\ev{A+2,B-2}}{3\ev{A+2,B-1}}\quad(B\geq2).
\]
For \(B=1\) the row-\(3\) outcome from \(\mu\) is unavailable and
\(p'_3=0\); for \(A=0\) the row-\(2\) outcome from \(\lambda\) has
probability zero and is deleted.  Cross-multiplying \(p_1\geq p'_1\) and
\(p_3\geq p'_3\) gives \eqref{eq:Lcut} and \eqref{eq:Rcut}.  The involution
\(\lambda\mapsto-w_0\lambda\) exchanges Up and Down steps and maps root
pairs to root pairs.  It therefore transports the Up coupling just proved
to the Down coupling for \((\lambda^*,\mu^*)\).
\end{proof}

\begin{rmk}[The case \(n=2\)]\label{rmk:rank2coupling}
For \(n=2\) the same reduction gives one condition.  For
\(\lambda=(m,0)\) and \(\mu=(m+1,-1)\) the admissible pairs are
\(1\mapsto\{1,2\}\) and \(2\mapsto\{2\}\), and the condition
\(p_2\leq p'_2\) reads \(\ev{m+1}/(2\ev{m})\geq\ev{m+3}/(2\ev{m+2})\) in
the notation of \eqref{eq:Deltam}, that is, \(r_{m+1}\geq r_{m+3}\).
Proposition~\ref{prop:logconcave} proves this monotonicity for all \(0<q,t<1\), so for
\(n=2\) the Up coupling exists for every root pair, and inversion gives
the Down coupling.  For \(n=3\) the inequalities \eqref{eq:Lcut} and
\eqref{eq:Rcut} are open (Remark~\ref{rmk:cuts-numeric}).
\end{rmk}

\begin{rmk}[Why rank-two log-concavity does not lift]\label{rmk:branching}
The evaluations \(\ev{A,B}\) are given by the branching rule as a
double sum over rank-two evaluations.  With \(w_m\) as in \eqref{eq:wk}
and \(g_m=(qt;q)_m/(t^2;q)_m\),
\begin{equation}
 \ev{A,B}=\sum_{i=0}^A\sum_{j=0}^B
 \frac{w_iw_{A-i}}{w_A}\,
 \frac{w_jw_{B-j}}{w_B}\,
 \frac{g_{B+i-j}\,g_{A+B}}{g_{A+B-j}\,g_{B+i}}\,
 \ev{B+i-j}.                                                     \label{eq:branch3}
\end{equation}
Up to factors depending on \(i\) alone and on \(j\) alone, the summand is
\(H_{B+i-j}\) with \(H_m=g_m\ev{m}\).  A family of kernels of the form
\(a(i)b(j)H(i-j)\), with positive \(a,b\), is multivariate totally positive
of order two (\(\mathrm{MTP}_2\)) on all its rectangles exactly when the
corresponding consecutive terms of \(H\) are log-concave
\citep[Chapter~3]{karlin1968}, and
\begin{equation}
 H_m^2-H_{m-1}H_{m+1}
 =g_m^2\bigl[\ev{m}^2-\omega_m\ev{m-1}\ev{m+1}\bigr],
 \qquad
 \omega_m-1=\frac{tq^{m-1}(1-q)(q-t)}{(1-q^mt)(1-q^mt^2)}.           \label{eq:barrier}
\end{equation}
For \(t<q\) one has \(\omega_m>1\), so the log-concavity of
Proposition~\ref{prop:logconcave} is not enough: the \(\mathrm{MTP}_2\)
property of the rank-three branching kernel is equivalent to
\begin{equation}
 \ev{m}^2\geq\omega_m\,\ev{m-1}\ev{m+1}\qquad(m\geq1).      \label{eq:strongbarrier}
\end{equation}
Inequality \eqref{eq:strongbarrier} is open; it holds in the tests of
Remark~\ref{rmk:cuts-numeric}, including where \(\omega_m>1\).  Whether it
implies \eqref{eq:Lcut}--\eqref{eq:Rcut} through \(\mathrm{MTP}_2\)
correlation inequalities is not settled either.
\end{rmk}

\begin{rmk}[Numerical evidence]\label{rmk:cuts-numeric}
The inequalities \(L_{A,B}\geq0\) and \(R_{A,B}\geq0\) of
Proposition~\ref{prop:hall} hold at all \(225\) points
\((q,t)\in\{i/16:1\leq i\leq15\}^2\) for the twelve pairs \((A,B)\) with
\(0\leq A\leq3\) and \(1\leq B\leq3\), and \eqref{eq:strongbarrier} holds at
the same points for \(m\leq7\), including where \(t<q\) and
\(\omega_m>1\).  These checks use exact rational arithmetic at each point;
they do not prove the inequalities for all \((q,t)\).  The identity
\eqref{eq:branch3} was checked exactly, as an identity in \((q,t)\), for
\(0\leq A\leq3\) and \(1\leq B\leq3\).
\end{rmk}

\section{The principal-specialization normalization fails}
\label{sec:omega}

Okounkov's \(q\)-integral formula \citep{okounkov1998} and the
finite-\(q\) lattice results of \citet{mcswiggen2026log,mcswiggen2026} are normalized at the
principal specialization rather than at \(1^n\).  This appendix shows that
the principal-specialization ratio does \emph{not} satisfy the analogue of
(M), and why.  Put
\[
 \Om_\lambda(x;q,t):=\frac{P_\lambda(x;q,t)}{P_\lambda(t^\delta;q,t)},
 \qquad t^\delta=(t^{n-1},t^{n-2},\ldots,t,1),
\]
and let
\[
 L_n^{q,t,a}
 :=\bigl\{\bigl(aq^{-\mu_1}t^{n-1},\,aq^{-\mu_2}t^{n-2},\ldots,aq^{-\mu_n}\bigr)
   :\mu_1\geq\cdots\geq\mu_n,\ \mu_i\in\Z\bigr\}
\]
be the scaled dominant \(q\)-lattice of \citet{mcswiggen2026log}, on which
Theorem~2.1 of that paper asserts that \(\lambda\mapsto\Om_\lambda(x;q,t)\)
is log-convex and Schur-convex.  Neither holds at the central point
\(1^2\), which lies on the lattice whenever \(t=q^k\) with
\(k\in\Z_{\geq1}\) (Lemma~\ref{lem:lattice}).

\subsection{The determinant shift acquires a factor}

Under the \(1^n\) normalization, \(P_\lambda(1^N)=P_{\lambda-c(1^N)}(1^N)\)
gives \(\F_\lambda^{(N)}(y)=(y_1\cdots y_N)^c\F_{\lambda-c(1^N)}^{(N)}(y)\)
for \(c=\lambda_N\).  Under the principal normalization the same shift
acquires a factor.

\begin{lemma}[The corrected shift]\label{lem:omega-shift}
Let \(\lambda\) have \(N\) parts and put \(c=\lambda_N\).  Then
\[
 \Om_\lambda^{(N)}(y;q,t)
 =\left(\frac{y_1\cdots y_N}{t^{N(N-1)/2}}\right)^{c}
  \Om_{\lambda-c(1^N)}^{(N)}(y;q,t).
\]
\end{lemma}

\begin{proof}
From \(P_\lambda(y)=(y_1\cdots y_N)^cP_{\lambda-c(1^N)}(y)\), evaluation at
\(t^\delta\) gives
\(P_\lambda(t^\delta)=t^{cN(N-1)/2}P_{\lambda-c(1^N)}(t^\delta)\); divide.
\end{proof}

For example, \(P_{(1,1)}(y_1,y_2)=y_1y_2\) and \(t^\delta=(t,1)\) give
\(\Om_{(1,1)}(y_1,y_2;q,t)=y_1y_2/t\), so the identity
\(\Om_{\lambda+(1^n)}=e_n\Om_\lambda\) fails for \(n=2\) unless \(t=1\).

\subsection{A counterexample for \texorpdfstring{\(n=2\)}{n=2}}

\begin{lemma}[When the central point lies on the lattice]\label{lem:lattice}
For \(n=2\), \((1,1)\in L_2^{q,t,a}\) for some \(a\) if and only if \(t=q^k\)
for an integer \(k\geq1\).  More generally, if \(t=q^k\) then
\(1^n\in L_n^{q,q^k,1}\), by \(\mu_i=k(n-i)\).
\end{lemma}

\begin{proof}
Matching coordinates, \(aq^{-\mu_2}=1\) forces \(a=q^{\mu_2}\), and then
\(aq^{-\mu_1}t=1\) forces \(t=q^{\mu_1-\mu_2}\) with \(\mu_1\geq\mu_2\);
\(0<t<1\) makes the exponent at least one.  Conversely
\(q^{-k(n-i)}t^{n-i}=1\) when \(t=q^k\).
\end{proof}

\begin{prop}[\(\Om\)-majorization fails on the lattice]\label{prop:omega-fail}
Let \(0<q,t<1\) and \(n=2\).  Then \((2,0)\maj(1,1)\), yet
\begin{equation}
 \label{eq:omegagap}
 \Om_{(1,1)}(1,1;q,t)-\Om_{(2,0)}(1,1;q,t)
 =\frac{(1-t)^2(1-qt)}{t(1+t)(1-qt^2)}>0 .
\end{equation}
If \(t=q^k\) with \(k\in\Z_{\geq1}\), then \((1,1)\in L_2^{q,t,1}\) by
Lemma~\ref{lem:lattice}, and \eqref{eq:omegagap} contradicts the
Schur-convexity of \(\lambda\mapsto\Om_\lambda(x;q,t)\) on the lattice.
\end{prop}

\begin{proof}
In two variables \(P_{(1,1)}(x_1,x_2)=x_1x_2\) and
\(P_{(2,0)}(x_1,x_2)=x_1^2+x_2^2+Ax_1x_2\) with
\(A=(1-t)(1+q)/(1-qt)\).  Hence \(\Om_{(1,1)}(1,1)=1/t\) and
\(\Om_{(2,0)}(1,1)=(2+A)/(1+t^2+At)\), so the difference is
\[
 \frac{1+t^2+At-t(2+A)}{t(1+t^2+At)}=\frac{(1-t)^2}{t(1+t^2+At)},
\]
the \(At\) terms cancelling.  Since
\(1+t^2+At=(1+t)(1-qt^2)/(1-qt)\), the difference is \eqref{eq:omegagap}, and it is
positive for \(0<q,t<1\).
\end{proof}

At \(q=t=r\) the polynomials are Schur, \((1,1)\in L_2^{r,r,1}\), and
\eqref{eq:omegagap} reduces to
\(1/r-3/(1+r+r^2)=(1-r)^2/\bigl(r(1+r+r^2)\bigr)\); at \(r=\nicefrac12\) the
two values are \(\Om_{(1,1)}=2\) and \(\Om_{(2,0)}=\nicefrac{12}{7}\).
Off the Schur line, at \(q=\nicefrac12\), \(t=q^2=\nicefrac14\), one has
\(A=\nicefrac97\) and \(\Om_{(2,0)}(1,1)=\nicefrac{368}{155}<4=\Om_{(1,1)}(1,1)\).

\begin{rmk}[Log-convexity fails as well]\label{rmk:omega-logconvex}
Along the one-row ray, \(\Om_{(0)}(1,1)=1\), \(\Om_{(1)}(1,1)=2/(1+t)\)
and \(\Om_{(2)}(1,1)=(2+A)/(1+t^2+At)\), and the same algebra gives
\begin{equation}
 \Om_{(1)}(1,1)^2-\Om_{(0)}(1,1)\,\Om_{(2)}(1,1)
 =\frac{(1-q)(1-t)^2}{(1+t)^2(1-qt^2)}>0 .                          \label{eq:omegalogconvex}
\end{equation}
So \(m\mapsto\Om_{(m)}(1,1;q,t)\) is not log-convex at \(m=1\), for any
\(0<q,t<1\).  In particular \(\lambda\mapsto\Om_\lambda(1^2;q,t)\)
cannot be a positive exponential sum \(\sum_vc_vq^{\ip{\lambda}{v}}\) in
\(\lambda\), since such sums are log-convex along every line.  At
\(q=t=\nicefrac12\) the three values are \(1\), \(\nicefrac43\), \(\nicefrac{12}7\),
and \((\nicefrac43)^2=\nicefrac{16}9>\nicefrac{12}7\).
\end{rmk}

\subsection{Why this leaves conjecture (M) untouched}

The anchored ratio satisfies \(\F_\lambda^{(n)}(1^n;q,t)=1\) for every
\(\lambda\), and in the smallest case
\[
 \F_{(2,0)}^{(2)}(x;q,t)-\F_{(1,1)}^{(2)}(x;q,t)
 =\frac{x_1^2+x_2^2+Ax_1x_2-(2+A)x_1x_2}{2+A}
 =\frac{(x_1-x_2)^2}{2+A}\ \geq\ 0 .
\]
So the anchored inequality holds, for every \(x\), in the case where the
\(\Om\)-inequality fails.  The difference is the factor
\(t^{-cN(N-1)/2}\) of Lemma~\ref{lem:omega-shift}, which the \(1^n\)
normalization cancels.  Nothing in this appendix bears on (M) itself; it
explains why the finite-\(q\) results for \(\Om\) do not transfer to (M).

\section{Relation to the work of McSwiggen and Sahi}\label{sec:audit}

Two recent preprints of McSwiggen and Sahi bear on (M); we state what they
prove and how it relates to the anchored problem.

\subsection{The Opdam--Cherednik kernel and the Jack case}

\citet{mcswiggen2026} prove (their Theorem~3.2) that for a finite root
system with nonnegative multiplicity \(k\) there is a probability measure
\(\pi_{k,x}\), supported on \(\conv(Wx)\), such that the nonsymmetric
Opdam--Cherednik kernel is its Laplace transform,
\begin{equation}
 G_{k,s-\rho}(x)
 =\int e^{\ip{s}{y}}\,\pi_{k,x}(dy).                              \label{eq:opdamlaplace}
\end{equation}
Symmetrizing over the Weyl group gives log-convexity of the Heckman--Opdam
hypergeometric function in its spectral variable and hence Weyl-majorization
inequalities for it (their Theorem~4.8), for all root systems and all real
\(x\).  In type \(A\), their Theorem~4.8 is the Schur-convexity of
\(s\mapsto\Fc_\theta(z,s)\) used in the proof of Theorem~\ref{thm:jack},
obtained from a different positive representation; combined with
\eqref{eq:spec} it gives the Jack inequality.  The paper's finite-\(q\)
statements (Proposition~4.3 and Theorem~4.4) concern the Macdonald
polynomial normalized as \(\mathtt P_\lambda=P_\lambda/P_\lambda(-\rho)\),
where evaluation of \(e^\nu\) at \(x\) means \(q^{\ip{\nu}{x}}\); in type
\(A\), \(\mathtt P_\lambda\) is a principal-specialization normalization, up to the central
shift and reversal of conventions.  They compare
\(\mathtt P_\lambda(\bar\mu)\) at shifted lattice points
\(\bar\mu=\mu-w_\mu^{-1}(\rho)\), using duality and the positive Ram--Yip
expansion, and they do not compare \(P_\lambda(X)/P_\lambda(1^n)\) for
arbitrary \(X\in\reals_{>0}^n\).  Their positive representation
\eqref{eq:opdamlaplace} lives at \(q=1\); it is not a finite-\(q\)
representation of the anchored ratio \eqref{eq:anchor}, and the Hankel
obstruction of Proposition~\ref{prop:hankel} does not bear on it.

\subsection{The lattice theorem of arXiv:2605.12680}

\citet{mcswiggen2026log} work with the principal-specialization
normalization \(\Om_\lambda\) of Appendix~\ref{sec:omega} and assert
(their Theorem~2.1) that for all \(q,t\in(0,1)\) and \(a>0\),
\(\lambda\mapsto\Om_\lambda(x;q,t)\) is log-convex and Schur-convex for
every \(x\) on the lattice \(L_n^{q,t,a}\).  Both assertions fail at the
central point: for \(t=q^k\), \(1^2\in L_2^{q,t,1}\)
(Lemma~\ref{lem:lattice}), and there \(\Om_{(2,0)}<\Om_{(1,1)}\)
(Proposition~\ref{prop:omega-fail}) while
\(\Om_{(1)}^2>\Om_{(0)}\Om_{(2)}\)
(Remark~\ref{rmk:omega-logconvex}).  Their Remark~2.2 notes, citing
\citet[Example~5.1]{chen2025}, that Schur-convexity of
\(\Om_\lambda(x)\) fails for some positive \(x\) off the lattice; the
point here is that both properties fail on the lattice itself.  The proof
of their Theorem~2.1 passes from the interlacing sum (their (2.1)) to a
positive exponential sum (their (2.3)) by the identity
\(\Om^{(N)}_{\lambda}(y)=(y_1\cdots y_N)^{\lambda_N}\,
\Om^{(N)}_{\lambda-\lambda_N(1^N)}(y)\), which by
Lemma~\ref{lem:omega-shift} holds only with the additional factor
\(t^{-\lambda_NN(N-1)/2}\); we have not traced every step at which the
argument fails at central points, but Remark~\ref{rmk:omega-logconvex}
rules out any representation of \(\lambda\mapsto\Om_\lambda(1^2)\) as
a positive exponential sum.

In that paper, Corollary~2.4 (the Jack inequality) and the implication
(1)\(\Rightarrow\)(2) of Theorem~2.5 are deduced from Theorem~2.1.  The
Jack statement is nonetheless correct: it is Theorem~\ref{thm:jack} here,
it follows from \citet[Theorem~4.8]{mcswiggen2026} through \eqref{eq:spec},
and it follows from Theorem~2.8 of \citet{mcswiggen2026log} itself, whose
proof does not use Theorem~2.1.  The counterexample does not bear on
\citet{mcswiggen2026}, whose type-\(A\) evaluation points \(\bar\nu\in W_0(\nu+\rho)\)
have pairwise distinct coordinates and are never central.

\begin{aiusage}  %
The results in this paper were developed through extensive use of an AI
assistant (OpenAI's GPT~Pro). None of the claims were obtained zero-shot:
they emerged through sustained back-and-forth discussion, including failed
attempts, counterexamples, audits, and repairs.
\end{aiusage}

\aiacknowledgements

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\end{document}