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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{permanent, van der Waerden, entropy, Alexandrov--Fenchel, Lorentzian, capacity, volumes}

\begin{document}
\title{A new proof of van der Waerden's inequality, and beyond}
\aiauthor

\begin{abstract}
  We give a new, self-contained proof of van der Waerden's permanent inequality via a monotone-averaging
  argument. Specifically, we show that the minimum total correlation of a random permutation with
  prescribed doubly stochastic marginals never decreases when two rows or two columns of the marginal
  matrix are averaged. This quantity is largest at the uniform matrix, which yields van der Waerden's
  bound. The monotone-averaging rests on the quadratic Alexandrov--Fenchel inequality, which lets us
  extend the argument to many other problems. In this way we also prove the Friedland--Tverberg
  inequalities and Gurvits's coefficient bound for Lorentzian polynomials.
\end{abstract}

\preprintslink{gurvits-lower}{31 October 2026}
\maketitle
\addtocontents{toc}{\protect\setcounter{tocdepth}{-1}}

\section{Introduction}\label{sec:intro}

Van der Waerden's inequality states that every doubly stochastic
\(n\times n\) matrix \(A\) satisfies \(\per A\ge n!/n^n\), with equality
only at the matrix \(J_n/n\) whose entries all equal \(1/n\).
\citet{Falikman1981} and \citet{Egorychev1981} proved it through the
structure of minimizers and the quadratic Alexandrov--Fenchel
inequality, and \citet{Gurvits2008} later gave a proof by capacity
induction for real stable polynomials.

The permanent itself need not decrease as \(A\) moves to \(J_n/n\) along
a straight segment \citep[Problem~8]{CheonWanless2005}, so a monotone
proof must follow a different quantity. We look for one that also
measures how far a matrix is from the extremal one, and whose
monotonicity rests on an inequality available beyond permanents.

We give a short, self-contained proof of van der Waerden's inequality
by a new monotonicity theorem. The quantity we follow is the minimum total correlation of a random
permutation \(\sigma\) with prescribed marginal matrix, that is, the
smallest value of \(\sum_i\Ent(\sigma(i))-\Ent(\sigma)\) over all such
\(\sigma\), where \(\Ent\) is Shannon entropy. This difference measures
how far the positions \(\sigma(1),\ldots,\sigma(n)\) are from
independent. We prove that the minimum never decreases when
two rows or two columns of the marginal matrix are averaged
(Theorem~\ref{chain:monotonicity}). Since finitely many averages reach
\(J_n/n\), it is largest there, and its maximum implies van der
Waerden's inequality and its equality case.

We show that the derivative along an average is nonnegative because
the permanent cofactors of the two averaged rows satisfy the quadratic
Alexandrov--Fenchel inequality, the same input that Falikman and
Egorychev used. Nothing else in the argument is specific to permanents, so we can apply
it to every form with the same quadratic property, including the mixed
derivatives of Lorentzian polynomials. At every step the argument keeps
a nonnegative remainder.

\vskip4pt
\noindent\textbf{Main contributions of the paper}.
\begin{enumerate}[label=(\arabic*),leftmargin=2.4em,labelsep=0.5em,itemsep=1pt,topsep=2pt]
\item A new monotonicity theorem (Theorem~\ref{chain:monotonicity}),
which shows that the minimum total correlation of a random permutation
with prescribed doubly stochastic marginals never decreases when two
rows or two columns of the marginal matrix are averaged; it gives a
short proof of van der Waerden's inequality with its equality case
(Corollary~\ref{chain:vdw}).
\item Averaging proofs of the Friedland--Tverberg inequalities and their
equality case (Theorem~\ref{thm:face-tverberg}) and of Gurvits's
coefficient bound for Lorentzian polynomials
(Corollary~\ref{cor:coefficient}). The latter gives lower bounds for
weighted counts of matroid bases and for mixed volumes, and hence for
the number of roots of generic polynomial systems
(Section~\ref{sec:applications}).
\item Remainders for these bounds. The change along each average splits
exactly into nonnegative entropy-production terms
(Theorem~\ref{thm:production}), and the bounds at every order give
explicit corrections; in particular, \(\per A\) exceeds \(n!/n^n\) by at
least a multiple of \(\|A-J_n/n\|_F^2\) whose constant depends only on
\(n\) and the smallest entry of \(A\) (Corollary~\ref{ft:explicit};
Proposition~\ref{md:remainder} for mixed discriminants).
\end{enumerate}

A second mechanism averages binary couplings toward independence and
gives the stable-pairing bound of Anari and Oveis Gharan with an exact entropy
remainder, which improves for two homogeneous polynomials of equal
degree (Theorem~\ref{thm:multiaffine-pairing} and
Proposition~\ref{prop:pairing-constraints}). For permanents it needs no
stability and corrects the Bethe lower bound, exactly on unions of even
cycles; for regular bipartite graphs it improves Schrijver's bound by a
factor exponential in the number of components
(Corollary~\ref{cor:bethe-correction}). Under a spectral gap, the
averaging remainders are at least a multiple of the squared distance to
equal rows (Corollary~\ref{cor:spectral-remainder}).

\subsection{Related work}\label{sec:related}
Table~\ref{tab:inequalities} lists the inequalities we prove, with the
earlier bound or input for each. The endpoints are known, and we credit
each where we use it; the averaging proofs and their remainders are
new. The entropy remainders are exact but implicit, since evaluating
them requires the maximum-entropy law, while those in item~(3), Proposition~\ref{prop:pairing-constraints},
and Corollary~\ref{cor:bethe-correction} are explicit.

\begin{table}[ht]
\centering
\footnotesize
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{@{}llll@{}}
\toprule
Setting & Inequality & Here & Earlier bound or input\\
\midrule
Permanents & \(\per A\ge n!/n^n\) & Cor.~\ref{chain:vdw} &
 \makecell[l]{\citet{Egorychev1981},\\ \citet{Falikman1981}}\\
Partial matchings & Friedland--Tverberg with remainder &
 Thm.~\ref{thm:face-tverberg} & \citet{Friedland1982}\\
Lorentzian \(f\) & \(f_\alpha\ge\binom d\alpha\alpha^\alpha d^{-d}\Cap_\alpha(f)\) &
 Cor.~\ref{cor:coefficient} & \citet{GurvitsNewton2009}\\
 & Product capacity with remainder &
 Cor.~\ref{cor:refined-lower} & \citet{GurvitsNewton2009}\\
 & Squarefree-coefficient sum & Cor.~\ref{cor:squarefree-sum} &
 \makecell[l]{\citet{AnariOveisGharanSaberiSingh2017}\\ (stable case)}\\
Mixed volumes & Volume-capacity lower bound &
 \eqref{eq:volume-lower} & \citet{GurvitsVolume2009}\\
Mixed discriminants & Determinant-capacity lower bound & \S\ref{sec:geometry} &
 \citet{Gurvits2008}\\
 & Explicit quadratic remainder &
 \makecell[l]{Prop.~\ref{md:remainder},\\ Cor.~\ref{ft:explicit}} &
 \makecell[l]{\citet{Gurvits2006},\\ \citet{FriedlandGurvits2006}}\\
Matroid bases & Ordered entropy with remainder &
 Cor.~\ref{cor:matroid} & \makecell[l]{Coefficient bound and\\ \citet{AnariOveisGharanVinzant2018}}\\
 & Weighted assignments; rainbow bases &
 \makecell[l]{Cor.~\ref{cor:matroid-assignments},\\ Cor.~\ref{cor:rainbow-bases}} & Same inputs\\
Stable pairings & Capacity bound with entropy remainder &
 Thm.~\ref{thm:multiaffine-pairing} & \citet{AnariOveisGharan2017}\\
Permanents & Bethe bound with explicit correction &
 Cor.~\ref{cor:bethe-correction} &
 \makecell[l]{\citet{Schrijver1998},\\ \citet{GurvitsSchrijver2012}}\\
\bottomrule
\end{tabular}
\caption{Inequalities proved by averaging.}
\label{tab:inequalities}
\end{table}

The closest antecedents of Theorem~\ref{chain:monotonicity} concern
different operations. \citet{Rothaus1972} and \citet{Bregman1973}
studied maximum-entropy permutation laws with prescribed marginals and
compared permanents through them. \citet{CryanGuoMousa2021} derive
entropy contraction from the same quadratic condition, and
\citet{LeeSeo2026} compare exchangeable Lorentzian laws under coordinate
deletion. Theorem~\ref{chain:monotonicity} instead compares the minimum
total correlation before and after each pair average. By
Remark~\ref{rem:capacity}, its endpoint at \(J_n/n\) is equivalent to
the capacity lower bound of \citet{Gurvits2008}, and the theorem adds monotonicity along the whole path.

\citet{AnariOveisGharan2017} proved the pairing bound for real stable
polynomials and derived Schrijver's inequality from it (their
Section~3.1). The variational form of the Bethe lower bound is due to
\citet[Theorem~2.2]{GurvitsSchrijver2012}, and \citet{Anari2026}
improves it by pairing rows. In Section~\ref{sec:schrijver-contraction}
we relate these results to ours.

The companion paper gives a rowwise upper bound on the maximum entropy
of a random permutation with prescribed marginals
\citep[Theorem~5.4]{sra2026bregman}. The supplement
\citep{sra2026supplement} transfers the Orlicz-norm bound of that paper
to mixed derivatives of polynomials and collects further consequences
of the averaging arguments.

\section{Pair averaging and van der Waerden's inequality}
\label{sec:permanent}\label{sec:maxent}

The monotone quantity comes from the entropy of random permutations. For
a doubly stochastic matrix \(P\in\Birk\), let \(\mathcal M(P)\) be the
set of probability laws on the symmetric group \(\Sn\) whose marginal
matrix is \(P\), that is, under which \(\sigma(i)=j\) with probability
\(p_{ij}\), and put
\begin{equation*}
 \mathsf H_{\mathrm{perm}}(P)=\max_{\mu\in\mathcal M(P)}\Ent(\mu),
 \qquad
 \mathcal S(P)=-\sum_{i,j}p_{ij}\log p_{ij},
 \qquad
 \mathcal I(P)=\mathcal S(P)-\mathsf H_{\mathrm{perm}}(P),
\end{equation*}
where \(\Ent\) denotes Shannon entropy, with natural logarithms and
\(0\log0=0\). Thus \(\mathcal I(P)\) is the
smallest total correlation that a random permutation with marginal
matrix \(P\) can have. Averaging rows \(r\) and \(s\) replaces \(P\) by
\(T_{rs}(t)P\), where \(T_{rs}(t)=(1-t)I_n+tS_{rs}\) and \(S_{rs}\)
interchanges rows \(r\) and \(s\).

\begin{dashedbox}
\begin{theorem}[Pair-averaging monotonicity]\label{chain:monotonicity}
For every \(P\in\Birk\), the function
\[
 t\longmapsto\mathcal I(T_{rs}(t)P)
\]
is nondecreasing on \([0,1/2]\). The same assertion holds for
right multiplication by a pair column-averaging matrix. Hence
\(\mathcal I\) is nondecreasing under every finite chain of pair
row and column \(T\)-transforms.
\end{theorem}
\end{dashedbox}

Both \(\mathcal S\) and \(\mathsf H_{\mathrm{perm}}\) are nondecreasing
along such averages, by concavity of entropy; for
\(\mathsf H_{\mathrm{perm}}\), mixing a maximizing law with its image
under the transposition of rows \(r\) and \(s\) shows this.
Theorem~\ref{chain:monotonicity} says that \(\mathcal S\) grows at least
as fast.

The monotonicity proof differentiates \(\mathcal I\) along an averaging
path, so it needs \(\mathcal I\) to be smooth with an explicit
differential. Both come from the Gibbs form of the maximizing law.
Maximum-entropy permutation laws with fixed marginals are classical
\citep[Theorem~2 and equation~(9)]{Bregman1973}; see also
\citet{Slater1987,Verducci1989,AgrawalWangYe2008}. We give the details needed here.

The quantity \(\mathcal I\) is well defined and nonnegative. Since
\(P\) has unit row and column sums, Hall's condition holds on its
positive support, and successively subtracting supported permutation
matrices with positive weights gives a law in \(\mathcal M(P)\).
Compactness and strict concavity give a unique entropy maximizer, and
entropy subadditivity gives \(\mathcal I(P)\ge0\). We write \(U=J_n/n\)
for the uniform matrix and
\(\mathcal V=\{X\in\R^{n\times n}:X\one=X^T\one=0\}\) for the directions
tangent to \(\Birk\), and for \(B>0\) we let
\(\mu_B(\sigma)=\prod_i b_{i,\sigma(i)}/\per B\) be the Gibbs law of
\(B\).

\begin{lemma}[Maximum-entropy parametrization]\label{chain:gibbs}
Every positive \(P\in\Birk\) is the marginal matrix of a positive
Gibbs law \(\mu_B\), which is its unique maximum-entropy law.
The matrix \(B\) can be chosen smoothly in \(P\) with \(B(U)=J_n\), and \(\mathsf H_{\mathrm{perm}}\) and \(\mathcal I\) are smooth
in the relative interior of \(\Birk\). For \(X\in\mathcal V\),
\begin{equation}\label{chain:differential}
 d\mathcal I(P)[X]
 =\sum_{i,j}X_{ij}\bigl(\log b_{ij}-\log p_{ij}\bigr).
\end{equation}
\end{lemma}

\begin{proof}
For \(P>0\), write \(P=\delta U+(1-\delta)R\) with
\(\delta>0\) and \(R\in\Birk\). Mixing a law for \(R\) with
the uniform law gives a positive law in \(\mathcal M(P)\).
An entropy maximizer cannot have a zero mass, since moving toward
this law would have infinite positive entropy derivative.
Lagrange multipliers therefore give \(\mu_B\), with
\(b_{ij}=e^{\theta_{ij}}\).
Conversely, relative entropy shows that every Gibbs law maximizes
entropy among laws with its marginals.

Put \(F(\theta)=\log\per(e^{\theta_{ij}})\), with entrywise
exponentiation. Its gradient is the marginal matrix and its Hessian
is the covariance of the permutation matrix. This covariance is
positive definite on \(\mathcal V\), because a constant statistic
\(\sum_iX_{i,\sigma(i)}\) has zero two-by-two alternating differences,
so \(X\) is a sum of row and column potentials, and hence is zero
if \(X\in\mathcal V\). Restricting \(\theta\) to \(\mathcal V\),
the inverse function theorem gives a smooth choice of \(B\), with
\(B(U)=J_n\). Differentiating
\(\mathsf H_{\mathrm{perm}}(P)=F(\theta)-\langle P,\theta\rangle\)
gives \(d\mathsf H_{\mathrm{perm}}(P)[X]=-\langle\theta,X\rangle\),
which proves \eqref{chain:differential}.
\end{proof}

\subsection{The local entropy comparison}

Along a pair-averaging path, \eqref{chain:differential} reduces to an
expression in a reversible stochastic matrix built from the cofactors
of the two selected rows. Two Jensen inequalities control its sign.

\begin{lemma}[Entropy and the quadratic spectral condition]\label{chain:kernel}
Let \(T\) be a reversible stochastic \(m\times m\) matrix with a
strictly positive stationary probability vector \(\pi\), and write
\(\langle u,v\rangle_\pi=\sum_j\pi_ju_jv_j\). Suppose
\begin{equation}\label{chain:negative-spectrum}
 \langle z,Tz\rangle_\pi
 \le \langle z,\one\rangle_\pi^2
 \qquad(z\in\R^m).
\end{equation}
Then, for every positive vector \(z\),
\begin{equation}\label{chain:kernel-entropy}
 \langle Tz-z,\log(Tz)\rangle_\pi\ge0.
\end{equation}
Equality holds exactly when \(Tz\) is constant.
\end{lemma}

\begin{proof}
Set \(a=\langle z,\one\rangle_\pi\). Jensen's inequality with weights
\(\pi_jz_j/a\) and then Jensen's inequality for \(s\log s\) give
\[
 \langle z,\log(Tz)\rangle_\pi
 \le a\log\frac{\langle z,Tz\rangle_\pi}{a}
 \le a\log a
 \le\langle Tz,\log(Tz)\rangle_\pi.
\]
Stationarity gives \(\langle Tz,\one\rangle_\pi=a\), so the last
inequality is strict unless \(Tz=a\one\). In that case all three
are equalities.
\end{proof}

The cofactor matrices of the permanent satisfy the hypothesis of
Lemma~\ref{chain:kernel} because of the quadratic Alexandrov--Fenchel
inequality \citep{Falikman1981,Egorychev1981}. We include a proof for the
particular cofactor matrices in use, so that no permanent lower bound is
assumed.

\begin{lemma}[Cofactor Alexandrov--Fenchel inequality]\label{chain:cofactor-af}
Let \(n\ge2\) and \(c_1,\ldots,c_{n-2}\in(0,\infty)^n\). Let \(M\) be the
symmetric matrix with diagonal zero and, for \(j\ne k\),
\[
 M_{jk}=\per\bigl((c_\ell)_{\ell=1}^{n-2}
                     \text{ with columns }j,k\text{ deleted}\bigr).
\]
For \(n=2\), the empty permanent is \(1\). Then
\begin{equation}\label{chain:cofactor-rcs}
 (x^TMy)^2\ge(x^TMx)(y^TMy)
 \qquad(x\in\R^n,\ y>0).
\end{equation}
In particular \(M\) has at most one positive eigenvalue.
\end{lemma}

\begin{proof}
A real polynomial is real stable if it has no zero when all variables
have positive imaginary parts. Positive directional differentiation
preserves stability for homogeneous polynomials of positive degree with
nonnegative coefficients. Indeed, if \(q\) is such a stable polynomial
and \(u>0\), all roots \(\alpha\) of \(q(z+tu)\) have
strictly negative imaginary parts when \(\operatorname{Im}z>0\).
Its leading coefficient is \(q(u)>0\), and
\(D_uq(z)/q(z)=\sum_\alpha(-\alpha)^{-1}\) has negative imaginary part.
Starting with \(z_1\cdots z_n\) and differentiating therefore gives
the real stable quadratic
\[
 q(z)=D_{c_1}\cdots D_{c_{n-2}}(z_1\cdots z_n)
     =\frac12z^TMz.
\]
For real \(x\) and positive \(y\), stability and conjugacy make every
root of \(q(x+ty)\) real. Its nonnegative discriminant is
\((x^TMy)^2-(x^TMx)(y^TMy)\), proving
\eqref{chain:cofactor-rcs}. The final assertion follows by restricting
the quadratic form to \(\{x:x^TMy=0\}\).
\end{proof}

\begin{proof}[\textbf{Proof of Theorem~\ref{chain:monotonicity}}]
Assume \(n\ge2\); the case \(n=1\) is immediate. For positive \(P\),
Lemma~\ref{chain:gibbs} supplies Gibbs parameters, and we put the
derivative along the averaging path into the form of
Lemma~\ref{chain:kernel}. Let \(B\) be the Gibbs matrix of \(P\), let
\(x,y\) be its two selected rows, and let \(M\) be the cofactor matrix of
the other rows from Lemma~\ref{chain:cofactor-af}. Define
\[
 C=\diag(y)M\diag(y),\quad \varrho=C\one,\quad
 T=\diag(\varrho)^{-1}C,\quad z=x/y,
 \quad Z=x^TMy=\varrho^Tz.
\]
All divisions here are coordinatewise. The matrix \(T\) is stochastic
and reversible for \(\pi=\varrho/(\one^T\varrho)\). Congruence and
\eqref{chain:cofactor-rcs} give
\(w^TCw\le(w^TC\one)^2/(\one^TC\one)\) for real \(w\), which normalizes
to \eqref{chain:negative-spectrum}.

Since \(\per B=x^TMy\), the corresponding rows \(u,v\) of \(P\) are
\[
 u_j=\frac{\varrho_jz_j}{Z},\qquad
 v_j=\frac{\varrho_j(Tz)_j}{Z},\qquad
 \log\frac{x_j}{u_j}-\log\frac{y_j}{v_j}=\log(Tz)_j.
\]
In the direction changing only these two rows,
\eqref{chain:differential} and Lemma~\ref{chain:kernel} give
\begin{equation}\label{chain:production}
 d\mathcal I(P)[(v-u,u-v)]=
 \frac1Z\sum_j\varrho_j\bigl((Tz)_j-z_j\bigr)\log(Tz)_j\ge0.
\end{equation}
At path parameter \(t<1/2\), the current row difference is \(1-2t\)
times the initial one, so the derivative along the original path is
\eqref{chain:production} divided by \(1-2t\); continuity gives the
endpoint \(t=1/2\).
Since \(T_{rs}(t)=S_{rs}T_{rs}(1-t)\) and \(\mathcal I\) does not change
when two rows are interchanged, \(\mathcal I(T_{rs}(t)P)\ge\mathcal I(P)\)
for every \(t\in[0,1]\), which gives the chain assertion.
Inversion of a random permutation shows
\(\mathsf H_{\mathrm{perm}}(P^T)=\mathsf H_{\mathrm{perm}}(P)\), while
\(\mathcal S(P^T)=\mathcal S(P)\), so the row assertion for \(P^T\) gives
the column assertion.

For boundary \(P\), set \(P_\varepsilon=(1-\varepsilon)P+\varepsilon U\).
Mixing feasible laws shows that \(\mathsf H_{\mathrm{perm}}\) is
concave, giving its lower radial limit. A convergent subsequence of
maximizing laws gives the matching upper limit by continuity of entropy.
Thus \(\mathcal I(P_\varepsilon)\to\mathcal I(P)\).
Since \(T_{rs}(t)U=U\), the same radial argument applies after averaging,
and passage to the limit proves the boundary statement.
\end{proof}

\subsection{The uniform endpoint}

Every doubly stochastic matrix reaches \(U\) through finitely many pair
averages, so Theorem~\ref{chain:monotonicity} makes \(U\) the maximizer
of \(\mathcal I\). The equality condition in Lemma~\ref{chain:kernel}
makes it the unique one.

\begin{theorem}[Unique uniform maximizer]\label{chain:maximum}
For \(n\ge2\),
\begin{equation*}
 0\le\mathcal I(P)\le\kappa_n:=n\log n-\log(n!),
 \qquad P\in\Birk.
\end{equation*}
Equality in the upper bound holds precisely at \(U=J_n/n\).
\end{theorem}

\begin{proof}
A finite chain of pair averages takes any list of rows to its mean.
Inductively average the first \(n-1\) rows to a common row \(u\),
and call the last row \(v\). Successively mix the last row with the
unused copies of \(u\), using parameters
\(1/n,1/(n-1),\ldots,1/2\). Each step fixes another row at
\(((n-1)u+v)/n\), and double stochasticity makes this row uniform.
Theorem~\ref{chain:monotonicity} therefore gives
\(\mathcal I(P)\le\mathcal I(U)=\kappa_n\).
Here the uniform law gives \(\mathsf H_{\mathrm{perm}}(U)=\log(n!)\),
while \(\mathcal S(U)=n\log n\).

For uniqueness, it suffices that the derivative
\eqref{chain:production} is positive near \(U\) whenever the two selected
rows differ. At \(U\), the smooth Gibbs choice in
Lemma~\ref{chain:gibbs} gives \(T=(J_n-I_n)/(n-1)\), which is
invertible. It stays invertible
near \(U\). By Lemma~\ref{chain:kernel}, zero production in
\eqref{chain:production} there forces \(Tz\) constant, hence \(z\)
constant and \(u=v\). On the last nontrivial segment of a chain
ending at \(U\), the selected rows are unequal before the endpoint.
Production is therefore strictly positive near that endpoint, giving
\(\mathcal I(P)<\mathcal I(U)\) whenever \(P\ne U\), including
boundary \(P\).
\end{proof}

\begin{corollary}[Van der Waerden's bound]\label{chain:vdw}
For every doubly stochastic \(A\),
\begin{equation*}
 \per A\ge\frac{n!}{n^n},
\end{equation*}
with equality precisely at \(J_n/n\).
\end{corollary}

\begin{proof}
Let \(\mu\) maximize entropy with marginals \(A\), and put
\(\nu(\sigma)=\prod_i a_{i,\sigma(i)}/\per A\). Hall's argument
gives \(\per A>0\), and \(\nu\) is positive wherever \(\mu\) is,
even with zero entries. Jensen's inequality gives
\begin{equation*}
 0\le\sum_\sigma\mu(\sigma)\log\frac{\mu(\sigma)}{\nu(\sigma)}
 =\log\per A+\mathcal I(A).
\end{equation*}
Theorem~\ref{chain:maximum} gives
\(\log\per A\ge-\mathcal I(A)\ge-\kappa_n\), with equality only
at \(U\). Direct evaluation gives \(\per U=n!/n^n\).
The case \(n=1\) is immediate.
\end{proof}

\begin{remark}[The spectral converse]\label{rem:kernel-converse}
For irreducible reversible \(T\), the entropy inequality
\eqref{chain:kernel-entropy} for every \(z>0\) implies
\eqref{chain:negative-spectrum}. Indeed, for \(z=\one+\varepsilon h\)
with \(\langle h,\one\rangle_\pi=0\), the left side is
\[
 \varepsilon^2\langle Th,(T-I)h\rangle_\pi+O(\varepsilon^3).
\]
On an eigenspace of eigenvalue \(\lambda\), the quadratic coefficient
is \(\lambda(\lambda-1)\|h\|_\pi^2\). Stochasticity gives
\(-1\le\lambda\le1\), and irreducibility makes \(1\) simple, so
universal nonnegativity excludes \(0<\lambda<1\).

Permanent kernels are irreducible because their off-diagonal entries
are positive. If we fix the other Gibbs rows and \(y>0\) and vary
\(x=y\odot z\), then \(z>0\) is arbitrary in \eqref{chain:production}. Thus pair-averaging monotonicity for every
interior marginal matrix recovers the cofactor Alexandrov--Fenchel
condition. The quadratic input to the local comparison is therefore
necessary as well as sufficient.
\end{remark}

\begin{remark}\label{rem:capacity}
The endpoint \(\mathcal I\le\kappa_n\) is equivalent to the capacity
lower bound
\[
 \per B\ge\frac{n!}{n^n}\inf_{y>0}
   \frac{\prod_i\sum_jb_{ij}y_j}{\prod_jy_j}
 \qquad(B>0)
\]
of \citet[Corollary~2.5 and Example~2.6]{Gurvits2008}; the permanent is
the case \(f=z_1\cdots z_n\) of the equivalence proved for Lorentzian
polynomials in Section~\ref{sec:capacity-remainders}. The endpoint alone
thus recovers Gurvits's bound, while Theorem~\ref{chain:monotonicity}
compares \(\mathcal I\) at successive points of every pair-averaging path.
\end{remark}

\section{Partial matchings by averaging within a face}
\label{sec:partial-matchings}

Van der Waerden's bound counts perfect matchings, and the
Friedland--Tverberg inequalities extend it to matchings of every size.
We prove them, with their equality case and an entropy remainder, by
averaging inside a face of a larger permutation polytope. For a
nonnegative \(n\times n\) matrix \(A\), let
\[
 \sigma_k(A)=\sum_{\substack{R,C\subseteq[n]\\|R|=|C|=k}}
                 \per A[R,C].
\]
For a bipartite adjacency matrix, this is the number of matchings of
size \(k\). \citet{Friedland1982} proved that \(\sigma_k\) is uniquely
minimized over doubly stochastic matrices at \(U=\one\one^T/n\) for
\(2\le k\le n\).
Friedland and Gurvits subsequently obtained a generalization for
positive hyperbolic polynomials \citep{FriedlandGurvits2006}.

To bring Theorem~\ref{chain:monotonicity} to bear, we embed matchings
into permutations. Applied to prescribed marginal matrices, the
completion construction of
\citet[Appendix~A, proof of Corollary~2.7]{AnariOveisGharanSaberiSingh2017}
turns a law on matchings of
size \(k\) into a permutation law on a fixed face of the permutation
polytope, and pair averages of \(A\) preserve that face.

The averaging argument then needs a functional on matchings that plays
the role of \(\mathcal I\). Fix \(2\le k\le n\). For a doubly stochastic
\(A\), let \(\mathsf H_k(A)\) be the maximum entropy of a law on size-\(k\) matchings whose edge marginals are
\((k/n)A\), and put
\[
 \mathcal F_k(A)=\frac{k}{n}\mathcal S(A)-\mathsf H_k(A).
\]
Selecting \(k\) edges uniformly from a random permutation with
marginals \(A\) gives such a law. Like \(\mathcal I\) for permutations,
\(\mathcal F_k\) compares the entropy of a random matching with the
entropy of its prescribed edge marginals.

\begin{theorem}[Averaging for partial matchings]
\label{thm:face-tverberg}
Let \(2\le k\le n\). The functional \(\mathcal F_k\) is nondecreasing
under pair averaging of the rows or columns of a doubly stochastic
matrix. Moreover,
\[
 \mathcal F_k(A)\le \mathcal F_k(U)=-\log\left(\binom nk^2\frac{k!}{n^k}\right),
\]
with equality if and only if \(A=U\). Consequently, writing
\(\Delta_{n,k}(A)=\mathcal F_k(U)-\mathcal F_k(A)\), one has
\begin{equation}\label{eq:face-tverberg}
 \sigma_k(A)\ge
 \binom nk^2\frac{k!}{n^k}\exp\bigl(\Delta_{n,k}(A)\bigr),
 \qquad \Delta_{n,k}(A)\ge0.
\end{equation}
Equality in the Friedland--Tverberg bound
\(\sigma_k(A)\ge\binom nk^2 k!/n^k\) holds exactly at \(U\).
\end{theorem}

\begin{proof}
\emph{\textbf{Completion.}} To apply Theorem~\ref{chain:monotonicity},
we complete each size-\(k\) matching to a permutation by adding
\(h=n-k\) rows and columns. Let \(N=n+h\), let \(J_{a,b}\) be the \(a\times b\)
all-ones matrix, and set
\[
 P_k(A)=
 \begin{pmatrix}
  (k/n)A&J_{n,h}/n\\
  J_{h,n}/n&0_{h,h}
 \end{pmatrix}.
\]
Blocks of size zero are omitted. This matrix is doubly stochastic.
The zero lower-right block restricts permutations to a set \(\Omega\).
Each allowed permutation uses exactly \(k\) upper-left entries,
and each size-\(k\) matching there has \((h!)^2\) completions.
Uniform conditional completion therefore adds \(2\log(h!)\) to
entropy. It also gives the prescribed marginals in the off-diagonal
blocks, because each original vertex is unmatched with probability
\(h/n\) and its added partner is uniform among \(h\) choices. The entropy
chain rule now gives
\[
 \mathsf H_{\mathrm{perm}}(P_k(A))=\mathsf H_k(A)+2\log(h!),\qquad
 \mathcal S(P_k(A))=\frac{k}{n}\mathcal S(A)-k\log(k/n)+2h\log n.
\]
Thus \(\mathcal I(P_k(A))-\mathcal F_k(A)\) is independent of \(A\).

\emph{\textbf{Monotonicity and the endpoint.}} A pair average of rows or columns of \(A\) acts on the corresponding
first \(n\) rows or columns of \(P_k(A)\). Applying
Theorem~\ref{chain:monotonicity} in dimension \(N\) proves monotonicity
of \(\mathcal F_k\), including at zero entries.
A finite chain takes the first \(n\) rows to their common mean,
with endpoint \(P_k(U)\). At \(U\), the uniform law on all
\(\binom nk^2k!\) matchings has the required edge marginals, so
\[
 \mathsf H_k(U)=\log\left(\binom nk^2k!\right),\qquad
 \frac{k}{n}\mathcal S(U)=k\log n.
\]
These values give \(\mathcal F_k(U)\), and the monotonicity gives
\(\Delta_{n,k}(A)\ge0\).

\emph{\textbf{The matching bound.}} The functional \(\mathcal F_k\)
controls \(\sigma_k\) through the Gibbs variational principle. Let
\(\mu\) maximize the matching entropy with edge marginals
\((k/n)A\). Every matching in its support uses only positive
entries of \(A\). Jensen's inequality therefore gives
\[
 \log\sigma_k(A)
 \ge \mathsf H_k(A)+\frac{k}{n}\sum_{i,j}a_{ij}\log a_{ij}
 =-\mathcal F_k(A).
\]
The same support argument applies at zero entries, and the feasible
law ensures \(\sigma_k(A)>0\). Substituting the definition of
\(\Delta_{n,k}\) proves \eqref{eq:face-tverberg}.

\emph{\textbf{Strictness.}} As in the proof of
Theorem~\ref{chain:maximum}, it suffices that the entropy derivative is
positive near the endpoint whenever the selected rows differ, so we
apply Lemma~\ref{chain:kernel} on the relative interior of the face with support \(\Omega\).
The uniform law on \(\Omega\) has marginal matrix \(P_k(U)\),
so this endpoint lies in that relative interior.
There the entropy maximizer has positive Gibbs parameters on all
allowed entries and zero parameters in the lower-right block.
These parameters can be chosen smoothly near \(P_k(U)\), with
value one on every allowed entry at the endpoint. Indeed, after
removing the affine relations on the incidence vectors, the Hessian
of the log partition function is a positive definite covariance
matrix, so the inverse function theorem applies.
For selected Gibbs rows \(x,y\) and cofactor matrix \(M\), the
kernel in the zero-production criterion is
\(T=\operatorname{diag}(C\one)^{-1}C\), with
\(C=\operatorname{diag}(y)M\operatorname{diag}(y)\) and \(z=x/y\)
coordinatewise.
At the endpoint \(x=y=\one\), this simplifies to
\(T=\operatorname{diag}(M\one)^{-1}M\).

Select two of the first \(n\) rows. At the endpoint, their two-slot
cofactor matrix \(M\) has zero diagonal and three off-diagonal values:
\[
 a=\frac{((n-2)!)^2}{(k-2)!},\qquad
 b=\frac{(n-1)!(n-2)!}{(k-1)!},\qquad
 c=\frac{n!(n-2)!}{k!}.
\]
They correspond respectively to two original columns, one original
and one added column, and two added columns; absent blocks are
omitted. To count the first case, assign the last \(h\) rows
injectively to the \(n-2\) remaining original columns, then assign
the remaining \(n-2\) rows bijectively to the remaining columns.
The count is \((n-2)!/(k-2)!\) times \((n-2)!\); the other two
cases replace \(n-2\) available original columns by \(n-1\) or
\(n\). The two row sums of \(M\) are
\[
 r_{\mathrm{o}}=\frac{((n-1)!)^2}{(k-1)!},\qquad
 r_{\mathrm{a}}=\frac{n!(n-1)!}{k!}.
\]
Consequently \(T=\operatorname{diag}(M\one)^{-1}M\) has
eigenvalue \(-(k-1)/(n-1)^2\) on the zero-sum vectors supported
on the \(n\) original columns. On the zero-sum vectors supported
on the \(h\) added columns its eigenvalue is \(-1/(n-1)\).
For \(h>0\), the remaining two-dimensional space of vectors
constant on each column class has quotient matrix
\[
 \frac1{n-1}
 \begin{pmatrix}k-1&h\\k&h-1\end{pmatrix},
\]
whose eigenvalues are \(1\) and \(-1/(n-1)\).
Thus the complete spectrum is
\begin{equation*}
 \left\{1,\quad
 -\frac{k-1}{(n-1)^2}\ \text{with multiplicity }n-1,\quad
 -\frac1{n-1}\ \text{with multiplicity }h\right\}.
\end{equation*}
For \(h=0\), the second group of nontrivial eigenvalues is absent.
In particular, the kernel is nonsingular for \(k\ge2\).

The kernel remains nonsingular near \(P_k(U)\). Approximating the zero
Gibbs parameters by positive ones shows that the cofactor
Alexandrov--Fenchel inequality remains valid on this face. Its
off-diagonal cofactors are positive for \(k\ge2\), so the kernel
and the entropy derivative are defined there. The zero-production
criterion in Lemma~\ref{chain:kernel} forces \(Tz\) constant, hence
\(z\) constant and the two selected marginal rows equal. If \(A\ne U\), the last
nonconstant segment of a finite averaging chain has unequal selected
rows before it reaches \(P_k(U)\). Near that endpoint all allowed
marginal entries are positive, even if the starting matrix is on the
boundary. Production is strictly positive on this part of the segment,
which proves \(\Delta_{n,k}(A)>0\). Direct evaluation at \(U\)
completes the equality statement.
\end{proof}

The remainder \(\Delta_{n,k}(A)\) is the integral of the permanent
entropy production along any chain averaging the first \(n\) rows
within this face, and Theorem~\ref{thm:production} below decomposes
that production into nonnegative local terms.
Corollary~\ref{ft:explicit} gives an explicit correction to
\(\sigma_k(A)\) in the entries of \(A\).

\section{Averaging for Lorentzian polynomials}\label{sec:lorentzian}

Mixed derivatives of Lorentzian polynomials are symmetric multilinear
forms whose two-slot contractions satisfy the quadratic
Alexandrov--Fenchel inequality, as the permanent does in
\eqref{chain:cofactor-rcs}. We extend Theorem~\ref{chain:monotonicity}
to them; at the uniform endpoint the extension becomes a capacity lower
bound for their coefficients.

To extend the argument, we replace random permutations by random words
whose weights are the coefficients of a polynomial. Throughout this
section, let
\[
 f(z)=\sum_{\alpha\in\Sf}f_\alpha z^\alpha,
 \qquad f_\alpha>0,\quad
 \Sf\subset\{\alpha\in\N^m:|\alpha|=d\},
\]
be a nonzero homogeneous polynomial of degree \(d\ge2\), and let
\(B\ge0\) have \(d\) rows \(b_1,\ldots,b_d\) and \(m\) columns. Its
mixed derivative is
\[
 \Df(B)=[t_1\cdots t_d]f(B^Tt)=D_{b_1}\cdots D_{b_d}f.
\]
For \(f=z_1\cdots z_n\) it is the permanent.

Expanding \(\Df(B)\) over words gives the analogue of the permanent's
sum over permutations. A word
\(\omega\in[m]^d\) has a multiplicity vector \(\alpha(\omega)\), and
we write \(\alpha!=\prod_j\alpha_j!\). Let \(\Omega_f=\{\omega\in[m]^d:\alpha(\omega)\in\Sf\}\).
For a word \(\omega\), its incidence matrix has a single one in
row \(i\), column \(\omega_i\). Expansion gives
\begin{equation*}
 \Df(B)=\sum_{\omega\in\Omega_f}w_\omega\prod_i b_{i,\omega_i},
 \qquad w_\omega=\alpha(\omega)!f_{\alpha(\omega)}.
\end{equation*}
The polytope \(\Kf\) is the convex hull of these incidence matrices.
A weighted entropy and its deficit then play the roles of
\(\mathsf H_{\mathrm{perm}}\) and \(\mathcal I\). For \(P\in\Kf\), with
rows \(p_i\), put
\[
 \Hf(P)=\max_{\mu:\,\mathrm{marg}\mu=P}
   \{\Ent(\mu)+\mathbb E_\mu\log w_\omega\},
 \qquad \If(P)=\mathcal S(P)-\Hf(P),
\]
where \(\mathcal S(P)=\sum_i\Ent(p_i)\) as before. For the permanent,
\(\Hf=\mathsf H_{\mathrm{perm}}\) and \(\If=\mathcal I\). In general
\(\If\) need not be nonnegative, because the coefficient weights need not
be probabilities. Pair averages of rows preserve the column sums
\(q=P^T\one\), so averaging ends at the matrix \(\bar P\) whose rows all
equal \(q/d\). The weighted entropy \(\Hf\) is concave. Its maximizing law exists by compactness and is unique by
strict concavity. For \(B>0\), Gibbs' variational formula is
\begin{equation}\label{eq:word-gibbs}
 \log\Df(B)=\max_{P\in\Kf}
       \{\langle P,\log B\rangle+\Hf(P)\}.
\end{equation}

The uniform endpoint \(\bar P\) is evaluated through capacity. Let
\(\Newt(g)\) denote the Newton polytope of a polynomial \(g\), the
convex hull of its exponent vectors.
For a polynomial \(g\) with nonnegative coefficients and an exponent
vector \(a\),
\begin{equation*}
 \Cap_a(g)=\inf_{z>0}\frac{g(z)}{z^a}.
\end{equation*}
If \(a\in\Newt(g)\), finite-dimensional convex duality gives
\begin{equation}\label{eq:occupation-dual}
 \log\Cap_a(g)=
 \max_{\nu:\,\mathbb E_\nu\alpha=a}
 \left\{\Ent(\nu)+\sum_\alpha\nu_\alpha\log g_\alpha\right\}.
\end{equation}
This entropy interpretation of capacity is standard
\citep[Appendix~A]{StraszakVishnoi2017}. The formula follows by applying the
log-sum-exp variational formula to \(g(e^u)\) and minimizing after
subtracting \(a\cdot u\). On a boundary face, the same argument applies
after restricting to that face, by compactness of the finite probability
simplex. In particular, the logarithm in \eqref{eq:occupation-dual} is
finite on \(\Newt(g)\).

\begin{proposition}[The equal-row entropy]\label{prop:equal-entropy}
For every nonnegative coefficient polynomial \(f\) and
\(q\in\Newt(f)\),
\begin{equation}\label{eq:equal-entropy}
 \Hf(\bar P)=\log(d!)+\log\Cap_q(f),
 \qquad \bar p_i=q/d.
\end{equation}
Moreover \(\Hf(P)\le\Hf(\bar P)\) for every \(P\in\Kf\)
with column sums \(q\).
\end{proposition}
\begin{proof}
We relax the marginal constraint to the expected multiplicity vector
\(q\). Symmetrizing a law over slot permutations preserves that
constraint and its expected log weight, does not decrease entropy,
and gives every slot marginal \(q/d\). Thus the relaxation is exact.
For a fixed count \(\alpha\), uniformity on its \(d!/\alpha!\)
words maximizes conditional entropy. The resulting objective for a
count law \(\nu\) is
\[
 \Ent(\nu)+\sum_\alpha\nu_\alpha\log(d!f_\alpha),
 \qquad \mathbb E_\nu\alpha=q.
\]
Formula \eqref{eq:occupation-dual} proves \eqref{eq:equal-entropy},
including boundary exponents. The same symmetrization starting from
a maximizing law for \(P\) proves the final assertion.
\end{proof}

Two established properties of homogeneous Lorentzian polynomials
enter the averaging argument. Their support is M-convex, and every
nonzero quadratic obtained from them by positive directional
differentiation has at most one positive eigenvalue. Nonnegative linear substitutions, directional differentiation, and
products preserve the class once the zero polynomial is admitted.
Homogeneous stable polynomials, volume polynomials, and matroid basis
generating polynomials belong to it
\citep[Proposition~2.2, Theorems~2.10, 2.25, 2.30, 3.10, and~4.1,
and Corollaries~2.11 and~2.32]{BrandenHuh2020}.

If we fix positive vectors in all but two slots of \(\Df\), the
remaining symmetric bilinear form is given by a matrix \(M\), the
two-slot contraction. If \(f\) is Lorentzian, this matrix is nonnegative and has at most one positive
eigenvalue. For positive \(y\), it follows that
\begin{equation}\label{eq:AF}
 (x^TMy)^2\ge(x^TMx)(y^TMy)\qquad(x\in\R^m).
\end{equation}
Indeed, the quadratic form is nonpositive on the \(M\)-orthogonal
complement of \(y\).
The value \(y^TMy\) is positive because \(f\ne0\) and the
contracting vectors are positive. Conversely this inequality implies
the one-positive-eigenvalue condition. Thus the kernel hypothesis
is exactly a quadratic Alexandrov--Fenchel hypothesis.

\begin{theorem}[Averaging for Lorentzian polynomials]\label{thm:averaging}
If \(f\) is Lorentzian and \(S_{rs}\) interchanges rows \(r,s\),
then
\[
 t\longmapsto\If\bigl(((1-t)I+tS_{rs})P\bigr)
\]
is nondecreasing for \(0\le t\le1/2\). In particular,
\begin{equation}\label{eq:rowcollapse}
 \If(P)\le\If(\bar P),
\end{equation}
and consequently the weighted entropy satisfies
\begin{equation}\label{eq:entropy-lower}
 \Hf(P)\ge\log\Cap_q(f)+\log(d!)+\mathcal S(P)-d\Ent(q/d).
\end{equation}
\end{theorem}

\begin{proof}
\emph{\textbf{Gibbs parameters.}} As for permanents, we first
parametrize interior marginal matrices by Gibbs laws. Let \(P_0\) be the
marginal matrix of the law proportional to \(w_\omega\), which is positive on \(\Omega_f\); by slot symmetry its
rows are identical. Every \(P\) in the relative interior of \(\Kf\) is
\((1-\varepsilon)P'+\varepsilon P_0\) for some \(P'\in\Kf\) and
\(\varepsilon>0\), so mixing a law for \(P'\) with this law gives a law
for \(P\) that is positive on \(\Omega_f\). The entropy maximizer must
also be positive, since moving towards a positive law has one-sided entropy derivative
\(+\infty\) at any zero coordinate. Lagrange
multipliers therefore give
\begin{equation}\label{eq:weighted-gibbs}
 \mu_B(\omega)=\frac{w_\omega\prod_i b_{i,\omega_i}}{\Df(B)},
 \qquad B>0.
\end{equation}
The Hessian of \(\log\Df(e^\theta)\) is the covariance of the
incidence vector. It is positive definite on the span of differences
of incidence vectors, because a zero-variance linear functional is
constant on the support and hence annihilates this span. After these
affine gauge directions are removed, the inverse function theorem proves
smoothness of \(\Hf\) in the relative interior and gives
\begin{equation}\label{eq:entropy-differential}
 d\If(P)[X]=\sum_{i,j}X_{ij}(\log b_{ij}-\log p_{ij})
\end{equation}
for tangent directions \(X\). Every active marginal coordinate is
positive there. The value is independent of the choice of Gibbs
parameters, because their nonuniqueness consists of affine relations
on the marginal polytope.

\emph{\textbf{The local comparison.}} In the relative interior we use the
Gibbs parameters \eqref{eq:weighted-gibbs} and bring the derivative into
the form of Lemma~\ref{chain:kernel}. Let the two selected parameter rows be
\(x,y\), and let \(M\) be the contraction of the other slots.
Define
\begin{equation}\label{eq:kernel-data}
 C=\diag(y)M\diag(y),\quad \varrho=C\one,\quad
 T=\diag(\varrho)^{-1}C,\quad \pi=\frac{\varrho}{\one^T\varrho},\quad z=x/y.
\end{equation}
Every \(\varrho_j\) is positive, because every retained coordinate occurs in the
word support, slot symmetry places it in the selected slot, and all
other parameter entries are positive. The matrix \(T\) is stochastic
and reversible for \(\pi\). Congruence and \eqref{eq:AF} give
\eqref{chain:negative-spectrum}. Write
\(a=\langle z,\one\rangle_\pi\). The two marginal rows are
\begin{equation}\label{eq:two-marginals}
 u_j=\frac{\pi_jz_j}{a},\qquad
 v_j=\frac{\pi_j(Tz)_j}{a}.
\end{equation}
Their gradient difference in \eqref{eq:entropy-differential} is
\(\log(Tz)_j\), so the swap-direction derivative is
\begin{equation}\label{eq:production-basic}
 d\If(P)[(v-u,u-v)]
 =\frac1a\langle Tz-z,\log(Tz)\rangle_\pi\ge0.
\end{equation}
At a point with path parameter \(t<1/2\), the original direction
is the current swap direction divided by \(1-2t\), which proves
monotonicity on the open path; continuity gives its endpoint.

\emph{\textbf{Boundary and endpoint.}} For boundary \(P\), we approximate
by \((1-\varepsilon)P+\varepsilon P_0\),
which lies in the relative interior. Pair averages fix \(P_0\), so they
act on this approximation through \(P\) alone. Concavity of the maximum
weighted entropy gives its lower radial limit. A convergent subsequence of maximizing laws gives the matching upper
limit, by continuity of entropy on the finite support. Passing to the limit proves the
boundary inequality.

The finite averaging construction in the proof of
Theorem~\ref{chain:maximum} takes any list of rows to their common mean.
Applied here, it proves \eqref{eq:rowcollapse}. Combining
\eqref{eq:rowcollapse} with Proposition~\ref{prop:equal-entropy} gives
\eqref{eq:entropy-lower}.
\end{proof}

Only the two-slot quadratic condition \eqref{eq:AF} was used in the
local comparison. For a nonnegative symmetric multilinear form,
requiring this condition at every positive choice of the other slots
is equivalent to its diagonal polynomial being Lorentzian
\citep[Proposition~2.33 and Theorem~2.30]{BrandenHuh2020}.

The monotonicity need not be strict. For \(f(z)=(a\cdot z)^d\) with
\(a>0\), we have \(\Hf(P)=\log(d!)+\mathcal S(P)+q\cdot\log a\), so
\(\If(P)=-\log(d!)-q\cdot\log a\) depends only on \(q\), and every pair
average leaves it unchanged.

Evaluating \eqref{eq:entropy-lower} at a deterministic word, we recover
Gurvits's coefficient bound.

\begin{corollary}[The Gurvits coefficient bound]\label{cor:coefficient}
For a degree-\(d\) Lorentzian polynomial and an integer
\(\alpha\in\N^m\) with \(|\alpha|=d\),
\begin{equation*}
 f_\alpha\ge\binom d\alpha\frac{\alpha^\alpha}{d^d}\Cap_\alpha(f),
 \qquad 0^0=1.
\end{equation*}
\end{corollary}
\begin{proof}
If \(\alpha\in\Sf\), take a deterministic word of that count.
Its marginal matrix has zero row entropy and weighted entropy
\(\log(\alpha!f_\alpha)\). The bound \eqref{eq:entropy-lower} gives the assertion. If
\(\alpha\notin\Sf\), M-convexity implies
\(\alpha\notin\Newt(f)\), so \(\Cap_\alpha(f)=0\) by a
separating hyperplane and the assertion is trivial.
\end{proof}

Corollary~\ref{cor:coefficient} is an entropy derivation of a theorem
of \citet[Corollary~3.6(2)]{GurvitsNewton2009}; see also
\citet[Theorem~2.10]{GurvitsLeake2026}.
For stable polynomials there are stronger degree-sensitive factors
\citep{Gurvits2008}. \citet[Theorem~1.4]{GurvitsLeake2026} prove that the
uniform polynomial is the unique minimizer of the all-ones coefficient
among doubly stochastic Lorentzian polynomials, by a capacity induction
that avoids the Alexandrov--Fenchel inequality.

\subsection{Entropy production}\label{sec:production}

The coefficient bound of Corollary~\ref{cor:coefficient} uses only the
endpoint of the averaging path. The derivative along the path is
nonnegative by Lemma~\ref{chain:kernel}, and its integral over a chain
from \(P\) to \(\bar P\) is the gain
\begin{equation}\label{eq:Delta-def}
 \Delta_f(P)=\If(\bar P)-\If(P)\ge0.
\end{equation}
Retaining the Jensen gaps in Lemma~\ref{chain:kernel} writes the
derivative as two relative entropies and the logarithmic slack in the
quadratic inequality. Their integral represents the gain as a
nonnegative remainder in a capacity lower bound.

\begin{theorem}[An exact production decomposition]\label{thm:production}
Assume that \(f\) is Lorentzian, or that the selected two-slot
contraction satisfies \eqref{eq:AF}. With the positive Gibbs data of
\eqref{eq:kernel-data} and the two marginal rows of
\eqref{eq:two-marginals}, set
\begin{equation*}
 R=\sum_j\frac{u_jv_j}{\pi_j},\qquad
 \eta_j=\frac{u_jv_j}{\pi_jR}.
\end{equation*}
Then \(0<R\le1\), \(\eta\) is a probability vector, and
\begin{equation}\label{eq:production-KL}
 d\If(P)[(v-u,u-v)]
 =\KL(v\Vert\pi)+\KL(u\Vert\eta)-\log R
 \ge\tfrac12\|v-\pi\|_1^2-\log R\ge0.
\end{equation}
Here \(\KL\) is relative entropy with natural logarithms.
\end{theorem}
\begin{proof}
The quadratic bound gives
\(R=\langle z,Tz\rangle_\pi/a^2\le1\).
Direct expansion yields
\[
 \KL(v\Vert\pi)=\sum_jv_j\log(Tz)_j-\log a,
 \quad
 \KL(u\Vert\eta)=\log(aR)-\sum_ju_j\log(Tz)_j.
\]
Subtracting \(\log R\) gives \eqref{eq:production-basic}.
Nonnegativity and Pinsker's inequality prove the estimates.
\end{proof}

As in the proof of Theorem~\ref{chain:monotonicity}, dividing
\eqref{eq:production-KL} by \(1-2t\) gives the derivative in the
original path direction at parameter \(t<1/2\). Consequently \(\Delta_f(P)\) is
the sum of these nonnegative production integrals along any finite
chain taking \(P\) to \(\bar P\). Integrals ending at \(1/2\)
are improper integrals with finite value by entropy continuity.
For boundary matrices the identity means the limit of the interior
production integrals under the radial approximation in the proof of
Theorem~\ref{thm:averaging}. The integrated value is independent of
the chain, since it is a difference of endpoint values.

Bounding an entropy gap by quadratic slack also appears in
\citet[proof of Lemma~10]{CryanGuoMousa2021}. Here the slack controls
production along prescribed marginal averages. A uniform negative
bound on the nontrivial kernel eigenvalues makes that control quantitative.

\begin{corollary}[Strictness and a spectral remainder]\label{cor:spectral-remainder}
Under the hypotheses of Theorem~\ref{thm:production}, put
\(a=\langle z,\one\rangle_\pi\). Then
\begin{equation}\label{eq:zero-production}
 d\If(P)[(v-u,u-v)]=0
 \quad\Longleftrightarrow\quad Tz=a\one
 \quad\Longleftrightarrow\quad M(x-ay)=0.
\end{equation}
If, for some \(0<\lambda\le1\),
\begin{equation}\label{eq:negative-gap}
 \langle h,Th\rangle_\pi\le-\lambda\|h\|_\pi^2
 \qquad\text{whenever }\langle h,\one\rangle_\pi=0,
\end{equation}
then, with \(c_\lambda=\lambda/(1+\lambda)^2\),
\begin{equation}\label{eq:spectral-production}
 d\If(P)[(v-u,u-v)]
 \ge c_\lambda\sum_j\frac{(u_j-v_j)^2}{\pi_j}.
\end{equation}
If \eqref{eq:negative-gap} holds with a common \(\lambda\) along a
finite pair-averaging chain in the relative interior of \(\Kf\)
from \(P\) to \(\bar P\), then
\begin{equation}\label{eq:spectral-stability}
 \Delta_f(P)\ge\frac{\lambda}{2(1+\lambda)^2}
                       \|P-\bar P\|_F^2.
\end{equation}
\end{corollary}
\begin{proof}
Lemma~\ref{chain:kernel} gives the first equivalence in
\eqref{eq:zero-production}; the definition of \(T\) in
\eqref{eq:kernel-data} gives the second. Set \(w=z/a-\one\).
Theorem~\ref{thm:production} gives
\[
 d\If(P)[(v-u,u-v)]\ge-\log R\ge1-R
                       =-\langle w,Tw\rangle_\pi.
\]
The nontrivial eigenvalues of the reversible stochastic matrix \(T\)
are \(-s\) with \(\lambda\le s\le1\). Since
\(s/(1+s)^2\ge c_\lambda\) on this interval,
\[
 -\langle w,Tw\rangle_\pi
 \ge c_\lambda\|(I-T)w\|_\pi^2
 =c_\lambda\sum_j\frac{(u_j-v_j)^2}{\pi_j},
\]
proving \eqref{eq:spectral-production}.

On a segment with initial rows \(u,v\) and parameter
\(0\le t\le\tau\le1/2\), the current row difference is
\((1-2t)(u-v)\). Dividing \eqref{eq:spectral-production} by
\(1-2t\), using \(\pi_j\le1\), and integrating gives a gain of
at least \(c_\lambda\tau(1-\tau)\|u-v\|_2^2\).
The squared Frobenius distance to \(\bar P\) decreases by exactly
\(2\tau(1-\tau)\|u-v\|_2^2\). Summing over the chain proves
\eqref{eq:spectral-stability}; continuity handles \(\tau=1/2\).
\end{proof}

Without such a gap the deficit can stay constant. For the power of a
linear form above, the nontrivial kernel eigenvalues are zero and
\(\If\) is constant at fixed column sums.

\subsection{Capacity bounds with remainders}
\label{sec:capacity-remainders}

To obtain a capacity bound with the entropy remainder \(\Delta_f\),
we evaluate \(\Hf\) at the matrix that optimizes
a transportation problem.
For a \(d\times m\) matrix \(B>0\), put
\(g_B(z)=\prod_i\langle b_i,z\rangle\). Rowwise Gibbs duality
gives
\begin{equation}\label{eq:transport-capacity}
 \log\Cap_q(g_B)=
 \max_{\substack{P\ge0,\ P\one=\one\\P^T\one=q}}
 \{\langle P,\log B\rangle+\mathcal S(P)\},\qquad |q|=d.
\end{equation}
For \(q=\one\) this is the formula of
\citet[Theorem~2.1]{StraszakVishnoiRelax2017}. The feasible
transportation polytope is compact, including when some \(q_j=0\).
The argument needs the optimizing matrix to be the marginal matrix of
a law on allowed words, and for M-convex supports the next lemma shows
that every matrix with unit row sums and column sums in \(\Newt(f)\) is
one.

\begin{lemma}[The marginal polytope]\label{lem:marginal-polytope}
If \(\Sf\) is M-convex, then
\begin{equation}\label{eq:marginal-polytope}
 \Kf=\{P\ge0:P\one=\one,\ P^T\one\in\Newt(f)\}.
\end{equation}
\end{lemma}
\begin{proof}
We realize the right side as a matroid-intersection polytope. The
integral-polymatroid description of an M-convex set gives the
increasing integer submodular function
\(r(J)=\max_{\alpha\in\Sf}\sum_{j\in J}\alpha_j\), with
\(r([m])=d\), and
\[
 \Newt(f)=\{q\ge0:q([m])=d,\ q(J)\le r(J)\ \forall J\},
 \qquad \Sf=\Newt(f)\cap\mathbb Z^m.
\]
On \(E=[d]\times[m]\), set \(E_J=[d]\times J\). The standard
matroid lift of an integer polymatroid \citep{Helgason1974} has rank
\[
 \widetilde r(F)=\min_{J\subseteq[m]}
                    \{r(J)+|F\setminus E_J|\}.
\]
By a theorem of \citet{McDiarmid1973}, in the form of
\citet[Theorem~2.1]{BoninChunFife2023}, its independent sets are exactly
the sets satisfying \(|F\cap E_J|\le r(J)\) for every \(J\).
We intersect this matroid with the partition matroid allowing at most
one element per row of \(E\).

If \(P\) belongs to the right side of \eqref{eq:marginal-polytope},
then \(p_{ij}\le1\), and for every \(F,J\),
\[
 P(F)\le P(E_J)+|F\setminus E_J|
       \le r(J)+|F\setminus E_J|.
\]
Thus \(P\) lies in both independent-set polytopes and has total
mass \(d\). Edmonds' matroid-intersection polytope theorem
\citep{Edmonds1979} expresses it as a convex combination of common
independent sets of size \(d\). Each chooses one element per row and
has its count vector in \(\Sf\); it is therefore an allowed word
incidence matrix. The reverse inclusion is immediate.
\end{proof}

The support hypothesis is needed. For \(f=z_1z_2+z_3z_4\), the
rows \((1/2,1/2,0,0)\) and \((0,0,1/2,1/2)\) have column sum
\((1/2,1/2,1/2,1/2)\in\Newt(f)\), but they force the two letters of
a word into different blocks, unlike every allowed word.

\begin{corollary}[Product-factor capacity and its entropy remainder]
\label{cor:refined-lower}
Let \(f\) be Lorentzian, \(q\in\Newt(f)\), and \(B>0\).
If \(P_q\) maximizes \eqref{eq:transport-capacity}, then
\begin{equation}\label{eq:refined-product-lower}
 \Df(B)\ge\frac{d!}{d^d}\,
 q^q\Cap_q(f)\Cap_q(g_B)\exp(\Delta_f(P_q)).
\end{equation}
In particular, dropping the remainder and taking the supremum over \(q\)
gives
\begin{equation}\label{eq:product-lower}
 \Df(B)\ge\frac{d!}{d^d}
 \sup_{q\in\Newt(f)}q^q\Cap_q(f)\Cap_q(g_B),
\end{equation}
also for \(B\ge0\).
\end{corollary}
\begin{proof}
Lemma~\ref{lem:marginal-polytope} makes every feasible \(P_q\)
a word marginal. Equations \eqref{eq:equal-entropy} and
\eqref{eq:Delta-def} give
\[
 \Hf(P_q)=\mathcal S(P_q)-d\Ent(q/d)+\log(d!)+\log\Cap_q(f)
                                      +\Delta_f(P_q).
\]
Inserting this law in \eqref{eq:word-gibbs} and using
\eqref{eq:transport-capacity} and \(\exp(-d\Ent(q/d))=q^q/d^d\) proves
\eqref{eq:refined-product-lower}. Dropping the nonnegative remainder
and taking a supremum proves \eqref{eq:product-lower}.
Approximation by \(B+\varepsilon\one\one^T\) proves the latter
for nonnegative matrices, since
\(\Cap_q(g_{B+\varepsilon\one\one^T})\downarrow\Cap_q(g_B)\) by
monotonicity and evaluation at a fixed positive point.
\end{proof}

Expanding the mixed derivative gives
\(\Df(B)/d!=\sum_{|\alpha|=d}\binom d\alpha^{-1}f_\alpha(g_B)_\alpha\),
so \eqref{eq:product-lower} is a Lorentzian analogue of the
coefficient-pairing conjecture of Gurvits
\citep[Conjecture~6.1]{GurvitsLeake2021} in the case of a product of
linear forms; the conjecture is stated there for two real stable
polynomials. In this case the unrefined
bound follows from the classical coefficient bound
\citep[Theorem~3.5(2)]{GurvitsNewton2009}. Indeed, we apply it to \(t\mapsto f(B^Tt)\), use
\(f(z)\ge\Cap_q(f)z^q\), and apply the transposed form of the
transportation duality \eqref{eq:transport-capacity}:
\[
 \inf_{t>0}\frac{\prod_j(\sum_i b_{ij}t_i)^{q_j}}{\prod_i t_i}
 =q^q\Cap_q(g_B).
\]
The log-sum-exp dual has objective
\(\langle P,\log B\rangle+\mathcal S(P)+\sum_jq_j\log q_j\)
with the same constraints as \eqref{eq:transport-capacity}; columns
with \(q_j=0\) are omitted.

The full entropy endpoint \eqref{eq:entropy-lower} is also a consequence
of this classical bound. Given \(P\) in the relative interior of \(\Kf\) with column sums \(q\),
we choose its Gibbs parameter matrix
\(B\). Combining
\eqref{eq:product-lower} with \eqref{eq:transport-capacity} gives
\[
 \Hf(P)=\log\Df(B)-\langle P,\log B\rangle
 \ge\mathcal S(P)+\log(d!)+\log\Cap_q(f)-d\Ent(q/d).
\]
Applying the radial continuity argument above to \(P\) and its
equal-row matrix gives the boundary cases. Together with the proof of
Corollary~\ref{cor:refined-lower}, this argument shows that the entropy endpoint
and the unrefined product-capacity bound are equivalent.

The corrected right side of \eqref{eq:refined-product-lower} equals
\[
 \exp\{\langle P_q,\log B\rangle+\Hf(P_q)\},
\]
which is the Gibbs variational bound at the transportation optimizer.
Pair averaging supplies an integral of nonnegative local terms for its
improvement over the capacity baseline. Positive production at any
interior point of a chain certifies a strict improvement.

For an explicit determinant-polynomial instance, take
\[
 f(z)=z_1^2+4z_1z_2+z_2^2,\qquad
 B=\begin{pmatrix}4&1\\1&4\end{pmatrix},\qquad q=(1,1).
\]
Then \(\Df(B)=84\),
\(f(B^Tt)=33t_1^2+84t_1t_2+33t_2^2\), and its capacity is
\(150\). Also \(\Cap_q(f)=6\), \(\Cap_q(g_B)=25\), and the
transport optimizer has rows \((4/5,1/5)\) and \((1/5,4/5)\).
Thus both unrefined lower bounds give \(75\).

Put \(s=(3+\sqrt{73})/4\), so \(2s^2-3s-8=0\).
The Gibbs matrix with rows \((s,1)\), \((1,s)\) has precisely
those marginals and partition function \(10(s+2)\). Consequently
\[
 \Hf(P_q)=\log(10(s+2))-\frac85\log s,
 \qquad
 \Delta_f(P_q)=\Hf(P_q)-2\Ent(4/5,1/5)-\log3.
\]
The refined bound is
\begin{equation*}
 84\ge10(s+2)(4/s)^{8/5}>75,
\end{equation*}
with middle value approximately \(82.3712\).
The strict improvement also has an exact production certificate:
the contraction is \(M=\left(\begin{smallmatrix}2&4\\4&2\end{smallmatrix}\right)\),
and at the starting point
\[
 R=\left(\frac{2s^2+8s+2}{4s^2+4s+4}\right)^2<1,
\]
since the denominator minus the numerator is \(2(s-1)^2>0\).
The initial production is therefore positive. Finally,
\(f(z)=\det(z_1\diag(2+\sqrt3,2-\sqrt3)+z_2I_2)\), so the example is also a mixed-discriminant instance of the correction,
with commuting matrices.

\section{Geometric and combinatorial inequalities}\label{sec:applications}

Theorem~\ref{thm:averaging} and its corollaries hold for every
Lorentzian polynomial. Geometry and combinatorics supply Lorentzian
polynomials whose mixed derivatives are mixed volumes, mixed
discriminants, intersection numbers, and weighted counts of matroid
bases, and for these the corollaries become inequalities of van der
Waerden type.

\subsection{Mixed volumes, mixed discriminants, and intersection numbers}
\label{sec:geometry}

Let \(K_1,\ldots,K_m\) be compact convex bodies in \(\R^d\),
and suppose their Minkowski sum is full dimensional. We use the
standard normalization \(V(K,\ldots,K)=\Vol(K)\). The volume polynomial is
\begin{equation}\label{eq:volume-polynomial}
 f(z)=\Vol\Bigl(\sum_jz_jK_j\Bigr)
 =\sum_{|\alpha|=d}\frac{d!}{\alpha!}
 V(K_1[\alpha_1],\ldots,K_m[\alpha_m])z^\alpha.
\end{equation}
For \(L_i=\sum_jb_{ij}K_j\), multilinearity gives
\begin{equation*}
 \Df(B)=d!V(L_1,\ldots,L_d).
\end{equation*}
The Lorentzian property follows from classical mixed-volume
Alexandrov--Fenchel theory \citep[Theorem~4.1]{BrandenHuh2020}.
All entropy comparisons and production identities therefore apply to
the coefficient weights in \eqref{eq:volume-polynomial}. In particular, applying Corollary~\ref{cor:coefficient} to
\(t\mapsto\Vol(\sum_it_iL_i)\) gives
\begin{equation}\label{eq:volume-lower}
 V(L_1,\ldots,L_d)
 \ge\frac1{d^d}\Cap_{\one}
       \left(\Vol\Bigl(\sum_i t_iL_i\Bigr)\right).
\end{equation}
Alternatively, for every \(q\in\Newt(f)\),
Corollary~\ref{cor:refined-lower} gives
\begin{equation}\label{eq:volume-product}
 V(L_1,\ldots,L_d)
 \ge\frac{q^q}{d^d}\Cap_q(f)\Cap_q(g_B),
\end{equation}
with the entropy correction of \eqref{eq:refined-product-lower} when
\(B>0\). Gurvits proved \eqref{eq:volume-lower}, together with stronger
factors involving the affine dimensions of the bodies
\citep[Theorems~2.1 and~2.4]{GurvitsVolume2009}, and
\eqref{eq:volume-product} follows from the classical coefficient bound
as in Section~\ref{sec:capacity-remainders}. When \(L_1,\ldots,L_d\)
are the Newton polytopes of a system of Laurent polynomials,
\(d!V(L_1,\ldots,L_d)\) is the number of isolated roots in
\((\mathbb C^*)^d\) of a generic system with these polytopes
\citep{Bernstein1975}, so \eqref{eq:volume-lower} bounds that generic
root count from below.

Let \(A_1,\ldots,A_m\) be Hermitian positive semidefinite matrices
of size \(d\), with \(\sum_j A_j\) positive definite, and put
\[
 f(z)=\det\Bigl(\sum_jz_jA_j\Bigr).
\]
This polynomial has nonnegative coefficients and is real stable
\citep[Proposition~3.2]{BorceaBrandenLiggett2009}, hence Lorentzian
\citep[Proposition~2.2]{BrandenHuh2020}.
We normalize the mixed discriminant by \(\MD(A,\ldots,A)=\det A\).
For \(Q_i=\sum_jb_{ij}A_j\),
\[
 \Df(B)=d!\MD(Q_1,\ldots,Q_d),\qquad
 f_\alpha=\frac{d!}{\alpha!}\MD(A_1[\alpha_1],\ldots,A_m[\alpha_m]).
\]
Equations \eqref{eq:volume-lower} and \eqref{eq:volume-product} hold
with \(V,\Vol\) replaced by \(\MD,\det\), with the same entropy
correction. The same applies to intersection numbers of nef divisor
classes on an irreducible projective variety, whose intersection
polynomial is Lorentzian \citep[Theorem~4.6]{BrandenHuh2020}.

\paragraph{Doubly stochastic tuples.}
\label{sec:mixed-stability}

Mixed discriminants of doubly stochastic tuples of matrices generalize
permanents of doubly stochastic matrices. For them, averaging gives the
partial bounds at every order, and keeping one slack term turns these
bounds into an explicit remainder in the matrices. For matrices of order
\(n\), we write \(\mathsf D=n!\MD\), so \(\mathsf D(I,\ldots,I)=n!\), and
we assume throughout that
\[
 A_i\succeq0,\qquad \operatorname{tr}A_i=1,
 \qquad \sum_{i=1}^n A_i=I.
\]
For \(0\le k\le n\), put
\[
 \mathsf S_k(A)=\sum_{\substack{S\subseteq[n]\\ |S|=k}}
       \mathsf D\bigl((A_i)_{i\in S},I^{[n-k]}\bigr),
 \qquad \mathsf S_0(A)=n!.
\]
Here \(I^{[r]}\) means \(r\) copies of \(I\). The following lower
bound on \(\mathsf S_k\) is a special case of the generalized
Friedland--Tverberg inequality of
\citet[Theorem~3.1]{FriedlandGurvits2006}, applied to the positive
hyperbolic polynomial \(q_k\) below; for \(k=n\) it is Gurvits's
solution of Bapat's conjecture \citep{Gurvits2006}. We include their
padding proof because Corollary~\ref{cor:coefficient} now supplies its
coefficient bound by averaging. We write
\(D_{\one}=\sum_i\partial_{x_i}\).

\begin{proposition}[Partial mixed-discriminant bound]\label{md:partial}
For every \(0\le k\le n\),
\begin{equation}\label{md:partial-bound}
 \mathsf S_k(A)\ge \binom nk\frac{n!}{n^k}.
\end{equation}
\end{proposition}

\begin{proof}
The cases \(k=0,1\) are identities. For \(k\ge2\), write
\[
 f(x)=\det\Bigl(\sum_i x_iA_i\Bigr),\qquad
 q_k(x)=\frac{k!}{n!}D_{\one}^{\,n-k}f(x),\qquad
 \ell(x)=\frac1n\sum_i x_i.
\]
The polynomial \(q_k\) is \(k\)-homogeneous. Since
\(f(\one)=1\) and \(\partial_i f(\one)=\operatorname{tr}A_i=1\),
homogeneity gives
\[
 q_k(\one)=1,\qquad
 \partial_iq_k(\one)=\frac kn.
\]
The determinant polynomial is stable, because if every \(z_i\) has positive
imaginary part, then the imaginary part of \(\sum_i z_iA_i\)
is positive definite, so this matrix is nonsingular.
It is therefore Lorentzian; nonnegative directional
differentiation and multiplication by a nonnegative linear form
preserve this class
\citep[Proposition~2.2, Corollaries~2.11 and~2.32, and
Theorem~2.30]{BrandenHuh2020}.
Thus
\(p_k=q_k\ell^{n-k}\) is Lorentzian, with
\(p_k(\one)=1\) and \(\partial_i p_k(\one)=1\).
Since its coefficients form a probability law on exponent vectors with
mean \(\one\), Jensen's inequality gives
\(p_k(x)\ge\prod_i x_i\) for \(x>0\). Consequently
\(\Cap_{\one}(p_k)=1\).

Corollary~\ref{cor:coefficient} therefore gives
\[
 [x_1\cdots x_n]p_k\ge\frac{n!}{n^n}.
\]
Only squarefree monomials of \(q_k\) can contribute on the left, so
\[
 [x_1\cdots x_n]p_k
 =\frac{k!}{n!}\frac{(n-k)!}{n^{n-k}}\mathsf S_k(A).
\]
Rearranging proves \eqref{md:partial-bound}.
\end{proof}

To express the remainder in the input matrices, define
\[
 \eta=n\min_i\lambda_{\min}(A_i),\qquad
 V(A)=\sum_i\|A_i-I/n\|_F^2
     =\sum_i\operatorname{tr}(A_i^2)-1.
\]
The assumptions imply \(0\le\eta\le1\).

\begin{proposition}[Explicit quadratic remainder]\label{md:remainder}
For \(n\ge2\) and \(2\le k\le n\),
\begin{equation}\label{md:remainder-bound}
 \mathsf S_k(A)\ge \binom nk\frac{n!}{n^k}
 +\binom{n-2}{k-2}
   \frac{(n-2)!}{2n^{k-2}}\eta^{k-2}V(A).
\end{equation}
The convention \(\eta^0=1\) applies also at \(\eta=0\).
In particular,
\begin{equation}\label{md:bapat-remainder}
 \mathsf D(A_1,\ldots,A_n)
 \ge\frac{n!}{n^n}
 +\frac{(n-2)!}{2n^{n-2}}\eta^{n-2}
       \sum_i\|A_i-I/n\|_F^2.
\end{equation}
For \(k=2\), \eqref{md:remainder-bound} is an identity. For every
\(k\ge2\), the coefficient of \(V(A)\) is sharp to second order at
\(A_i=I/n\).
\end{proposition}

\begin{proof}
If \(\eta=1\), all \(A_i=I/n\), so the assertion is immediate.
Otherwise we split off the uniform part \((\eta/n)I\) of each \(A_i\);
what remains is again a doubly stochastic tuple, and \(\mathsf S_k(A)\)
expands into its slacks. We set
\[
 B_i=\frac{A_i-\eta I/n}{1-\eta}.
\]
Multilinearity and
counting the sets that contain a fixed \(j\)-element subset give
\begin{equation}\label{md:expansion}
 \mathsf S_k(A)=\sum_{j=0}^k \binom{n-j}{k-j}
   \left(\frac\eta n\right)^{k-j}(1-\eta)^j\mathsf S_j(B).
\end{equation}
Set
\(\delta_j(B)=\mathsf S_j(B)-\binom njn!/n^j\).
Proposition~\ref{md:partial} says \(\delta_j(B)\ge0\), and the
binomial identity
\(\binom{n-j}{k-j}\binom nj=\binom nk\binom kj\)
shows that \eqref{md:expansion} is exactly
\begin{equation}\label{md:slack-expansion}
 \mathsf S_k(A)-\binom nk\frac{n!}{n^k}
 =\sum_{j=2}^k \binom{n-j}{k-j}
   \left(\frac\eta n\right)^{k-j}(1-\eta)^j\delta_j(B).
\end{equation}

The order-two term is explicit. Differentiating the determinant twice
at the identity gives
\[
 \mathsf D(B_i,B_j,I^{[n-2]})
 =(n-2)!\bigl(1-\operatorname{tr}(B_iB_j)\bigr).
\]
Since \(\sum_iB_i=I\), summing over \(i<j\) yields
\[
 \delta_2(B)=\frac{(n-2)!}{2}
       \left(\sum_i\operatorname{tr}(B_i^2)-1\right)
 =\frac{(n-2)!}{2}V(B).
\]
Keeping only \(j=2\) in \eqref{md:slack-expansion} and using
\((1-\eta)^2V(B)=V(A)\) proves the bound. At \(k=2\) no terms
were discarded.

For the second-order assertion, let
\(A_i(\varepsilon)=I/n+\varepsilon H_i\), where the \(H_i\) are
Hermitian, \(\operatorname{tr}H_i=0\), and \(\sum_iH_i=0\).
Multilinearity gives
\[
 \mathsf S_k(A(\varepsilon))=\binom nk\frac{n!}{n^k}
 +\binom{n-2}{k-2}\frac{(n-2)!}{2n^{k-2}}
       \varepsilon^2\sum_i\operatorname{tr}(H_i^2)
 +O(\varepsilon^3).
\]
Indeed, the linear term vanishes; the quadratic term uses
\(\mathsf D(H_i,H_j,I^{[n-2]})=-(n-2)!\operatorname{tr}(H_iH_j)\)
and \(\sum_{i<j}\operatorname{tr}(H_iH_j)
=-\frac12\sum_i\operatorname{tr}(H_i^2)\).
Also \(\eta(A(\varepsilon))=1+O(|\varepsilon|)\), proving the
asserted local sharpness.
\end{proof}

For positive definite tuples, \(\eta>0\), so
\eqref{md:bapat-remainder} is strict unless every \(A_i=I/n\).
For singular tuples and \(n>2\), its displayed correction vanishes;
it does not by itself settle boundary equality. At \(n=2\), it is the
exact identity
\(\mathsf D(A_1,A_2)=\tfrac12+\tfrac12V(A)\).

If \(A_i=\operatorname{diag}(a_{i1},\ldots,a_{in})\), the tuple
conditions say that the matrix with entries \(a_{ij}\) is doubly
stochastic, and \(\mathsf S_k(A)\) is \((n-k)!\) times its
\(\sigma_k\) from Section~\ref{sec:partial-matchings}. Proposition~\ref{md:partial} then
recovers the Friedland--Tverberg constant \(\binom nk^2 k!/n^k\), and
Proposition~\ref{md:remainder} corrects it in the entries of the matrix.

\begin{corollary}[Explicit partial-permanent remainder]\label{ft:explicit}
Let \(A\) be an \(n\times n\) doubly stochastic matrix,
\(a=\min_{i,j}a_{ij}\), and \(2\le k\le n\). Then, with
\(b_{n,k}=\binom nk^2 k!/n^k\),
\begin{equation}\label{ft:explicit-bound}
 \sigma_k(A)\ge b_{n,k}
 +\frac12\binom{n-2}{k-2}^{\!2}(k-2)!\,a^{k-2}
          \|A-U\|_F^2.
\end{equation}
Here \(a^0=1\), including when \(a=0\). The bound is an identity
for \(k=2\), and its quadratic coefficient is asymptotically exact
as \(A\to U\).
\end{corollary}

\begin{proof}
We apply Proposition~\ref{md:remainder} to the diagonal matrices whose
diagonals are the rows of \(A\). Then \(\eta=na\),
\(V=\|A-U\|_F^2\), and \(\mathsf S_k=(n-k)!\,\sigma_k(A)\). Dividing by
\((n-k)!\) and using \((n-2)!/(n-k)!=\binom{n-2}{k-2}(k-2)!\) gives
\eqref{ft:explicit-bound}. The identity at \(k=2\) and the local
sharpness are those of Proposition~\ref{md:remainder}, since diagonal
perturbations are allowed there.
\end{proof}

For example, for the matrix with diagonal entries \(1/2\) and
off-diagonal entries \(1/4\) in dimension three, the permanent is
\(1/4\). The lower bound in \eqref{ft:explicit-bound} is
\(2/9+1/64=137/576\), compared with the unrefined value \(2/9\).
For other quantitative permanent estimates, see
\citet[Theorem~1.2]{Lavi2026}.

\subsection{Matroids and strongly Rayleigh laws}
\label{sec:matroids}

For a multiaffine polynomial, the mixed derivative is a weighted count
of assignments of labelled slots to bases. Let
\(f(z)=\sum_{S\in\mathcal B}f_Sz^S\) be homogeneous and multiaffine of
degree \(d\). Then
\[
 \Df(B)=\sum_{S\in\mathcal B}f_S\per(B_S),
\]
the weighted partition function for assigning \(d\) labelled slots
bijectively to a chosen set \(S\); if \(f(\one)=1\), it is
\(\mathbb E_S\per(B_S)\). The averaging theorems need \(f\) to be
Lorentzian, which arbitrary positive basis weights need not be. A
fixed-size strongly Rayleigh distribution has a homogeneous stable
generating polynomial and is therefore covered. A general strongly
Rayleigh law may first be conditioned on a cardinality with positive
probability, which preserves the strongly Rayleigh property
\citep[Corollary~4.18]{BorceaBrandenLiggett2009}. Every unweighted
matroid basis polynomial is Lorentzian, including those of matroids
without a stable basis polynomial
\citep[Theorem~3.10]{BrandenHuh2020}. Positive external fields preserve
this property.

For such \(f\), \eqref{eq:entropy-lower} bounds the maximum weighted
entropy of ordered bases with prescribed slot marginals. For an
unweighted matroid, the entropy estimate of
\citet[Theorem~5.2, Corollary~5.6, and
Lemma~5.8]{AnariOveisGharanVinzant2018} applies to both a basis law and
its complement. Keeping both entropy estimates strengthens the bound
for ordered bases; to state it, we write
\[
 L(q)=-\sum_jq_j\log q_j,\qquad
 \chi(q)=\max\{0,L(\one-q)-L(q)\},\qquad
 \kappa_d=\log\frac{d^d}{d!}.
\]

\begin{corollary}[Ordered matroid bases]\label{cor:matroid}
Let \(f\) be the unweighted basis generating polynomial of a matroid
of rank \(d\) on \([m]\), and let \(P\in\Kf\) have column sums \(q\).
Then
\begin{equation}\label{eq:matroid-ordered-refined}
 \Hf(P)\ge\mathcal S(P)-\kappa_d+\chi(q)+\Delta_f(P).
\end{equation}
\end{corollary}

\begin{proof}
The prescribed-marginal construction of
\citet[Lemma~5.8]{AnariOveisGharanVinzant2018} gives a basis law
\(\nu\) with marginals \(q\) such that \(\nu\) and its complement
\(\nu^*\) are log-concave. Their entropy estimate gives
\(\Ent(\nu)\ge L(q)\) and
\(\Ent(\nu)=\Ent(\nu^*)\ge L(\one-q)\). The construction includes
boundary marginals by limits of external fields. By
\eqref{eq:occupation-dual},
\begin{equation}\label{eq:matroid-capacity-entropy}
 \log\Cap_q(f)\ge\max\{L(q),L(\one-q)\}=L(q)+\chi(q).
\end{equation}
Equations \eqref{eq:equal-entropy} and \eqref{eq:Delta-def} give
\[
 \Hf(P)=\mathcal S(P)-\kappa_d-L(q)
              +\log\Cap_q(f)+\Delta_f(P),
\]
which proves the claim.
\end{proof}

For the free matroid of rank \(n\), the polynomial is
\(z_1\cdots z_n\), the column sums are \(q=\one\), and
\(\chi(q)=0\). Dropping \(\Delta_f(P)\) recovers the bound
\(\mathcal I\le\kappa_n\) of Theorem~\ref{chain:maximum}.

External fields carry these bounds to weighted bases. For positive
external fields \(\lambda\), put
\(f_\lambda(z)=f(\lambda_1z_1,\ldots,\lambda_mz_m)\). Then
\[
 \mathsf H_{f_\lambda}(P)=\Hf(P)+q\cdot\log\lambda,
 \qquad
 \Cap_q(f_\lambda)=\lambda^q\Cap_q(f),
 \qquad
 \Delta_{f_\lambda}(P)=\Delta_f(P).
\]
For the normalized polynomial \(f_\lambda/f(\lambda)\), the
normalization subtracts \(\log f(\lambda)\) from both the entropy and the
logarithmic capacity and leaves the averaging remainder unchanged. These identities hold
on the boundary as well, since the log weight added to every admissible
law has expectation \(q\cdot\log\lambda\).

The ordered entropy bound also compares a weighted assignment count
with a relaxation involving only the slot marginals. For a
\(d\times m\) matrix \(B\ge0\), let
\[
 \mathcal P_M(B)=\{P\in\Kf:p_{ij}=0\text{ whenever }b_{ij}=0\},
 \qquad
 F_B(P)=\langle P,\log B\rangle+\mathcal S(P),
\]
where zero terms are omitted in the inner product. If
\(\mathcal P_M(B)\ne\varnothing\), set
\[
 Z_M(B)=\sum_{S\text{ basis of }M}\per(B_S),\qquad
 \rho_M(B)=\max_{P\in\mathcal P_M(B)}F_B(P).
\]

\begin{corollary}[Weighted assignments]\label{cor:matroid-assignments}
Let \(P\in\mathcal P_M(B)\) have column sums \(q\). Then
\begin{equation}\label{eq:matroid-assignment-refined}
 Z_M(B)\ge\frac{d!}{d^d}
       \exp\bigl(F_B(P)+\chi(q)+\Delta_f(P)\bigr).
\end{equation}
Consequently, the relaxation value \(\rho_M(B)\) satisfies
\begin{equation}\label{eq:matroid-assignment-relaxation}
 \frac{d!}{d^d}\,e^{\rho_M(B)}
       \le Z_M(B)\le e^{\rho_M(B)}.
\end{equation}
Moreover, with \(g_B(z)=\prod_i\langle b_i,z\rangle\) as in
Section~\ref{sec:capacity-remainders},
\[
 e^{\rho_M(B)}=\sup_{q\in\Newt(f)}\Cap_q(g_B).
\]
\end{corollary}

\begin{proof}
The Gibbs formula \eqref{eq:word-gibbs} gives
\[
 \log Z_M(B)=\max_{P\in\mathcal P_M(B)}
       \{\langle P,\log B\rangle+\Hf(P)\}.
\]
For each feasible \(P\), its maximum-entropy law is supported on
allowed assignments, because a zero marginal excludes that coordinate from
every word of positive probability. Applying
\eqref{eq:matroid-ordered-refined} proves
\eqref{eq:matroid-assignment-refined}; applying entropy subadditivity
\(\Hf(P)\le\mathcal S(P)\) proves the upper bound in
\eqref{eq:matroid-assignment-relaxation}. The lower bound follows by
choosing a maximizer of \(F_B\) and dropping the two nonnegative
corrections. Finally, Lemma~\ref{lem:marginal-polytope} lets us optimize
first at fixed column sums, and \eqref{eq:transport-capacity} then gives
the last identity. The same variational formula holds for
zero entries of \(B\) after restricting the support.
\end{proof}

The factor \(d!/d^d\) itself also follows from the classical
coefficient bound and \eqref{eq:matroid-capacity-entropy}, through
Corollary~\ref{cor:refined-lower}; the averaging adds the entropy
remainder at the chosen marginals. Corollary~\ref{cor:matroid-assignments}
thus specializes the entropy approach to matroid counting of
\citet{AnariOveisGharanVinzant2018} to assignments with prescribed
slot marginals.

For disjoint color classes \(E_1,\ldots,E_d\) partitioning \([m]\),
an assignment is a rainbow basis when slot \(i\) must choose from
\(E_i\). A vector \(q\in\Newt(f)\) with \(q(E_i)=1\) determines
the feasible matrix
\(P(q)_{ij}=q_j\mathbf1_{\{j\in E_i\}}\).

\begin{corollary}[Rainbow bases]\label{cor:rainbow-bases}
Let \(M\) be a rank-\(d\) matroid, with the color classes above and
positive element weights \(\lambda_j\). For every
\(q\in\Newt(f)\) satisfying \(q(E_i)=1\),
\begin{equation*}
 \sum_{\substack{S\text{ basis of }M\\|S\cap E_i|=1\ (1\le i\le d)}}
       \prod_{j\in S}\lambda_j
 \ge\frac{d!}{d^d}
       \exp\bigl(q\cdot\log\lambda
             +\max\{L(q),L(\one-q)\}+\Delta_f(P(q))\bigr).
\end{equation*}
\end{corollary}

\begin{proof}
We apply \eqref{eq:matroid-assignment-refined} to
\(b_{ij}=\lambda_j\mathbf1_{\{j\in E_i\}}\).
Each rainbow basis has a unique allowed assignment, and
\(F_B(P(q))=q\cdot\log\lambda+L(q)\).
\end{proof}

The supplement specializes these bounds to weighted rainbow spanning
trees and to the homogenized multivariate Tutte polynomial
\citep[Section~S5.2]{sra2026supplement}.

\subsection{Sums of squarefree coefficients}
\label{sec:squarefree-sums}

\citet[Theorem~1.2]{AnariOveisGharanSaberiSingh2017} bound a squarefree
coefficient sum in their analysis of Nash social welfare, and this bound
also admits an averaging proof. Their padding
construction \citep[Lemma~4.1]{AnariOveisGharanSaberiSingh2017} reduces
the sum to a single coefficient. Applying
Corollary~\ref{cor:coefficient} to the padded polynomial gives their
constant throughout the Lorentzian class.

For a homogeneous polynomial \(f\) of degree \(d\le m\), put
\[
 C_{\mathrm{sf}}(f)=\sum_{\substack{S\subseteq[m]\\|S|=d}}f_{\one_S},
 \qquad
 \mathcal R(f)=\inf_{\substack{z>0\\z^S\ge1\ (|S|=d)}}f(z),
 \qquad
 \mathcal Q_{m,d}=\{q\in[0,1]^m:|q|=d\}.
\]
As usual, a product with exponent zero is interpreted as one.

\begin{corollary}[Squarefree coefficient sums]
\label{cor:squarefree-sum}
Let \(f\) be a nonzero Lorentzian polynomial of degree \(1\le d\le m\)
with \(C_{\mathrm{sf}}(f)>0\). Set \(h=m-d\) and
\(c_{m,d}=m!h^h/(m^m h!)\). Then
\begin{equation}\label{eq:squarefree-sum}
 C_{\mathrm{sf}}(f)
 \ge c_{m,d}\sup_{q\in\mathcal Q_{m,d}}
       \frac{\Cap_q(f)}{(\one-q)^{\one-q}}
 \ge c_{m,d}\mathcal R(f)
 \ge e^{-d}\mathcal R(f).
\end{equation}
\end{corollary}
\begin{proof}
The case \(m=1\) is an identity, so assume \(m\ge2\).
The product
\(F(z)=f(z)(z_1+\cdots+z_m)^h\) is Lorentzian by
\citet[Corollary~2.32 and Theorem~2.30]{BrandenHuh2020}, and
\[
 [z_1\cdots z_m]F=h!C_{\mathrm{sf}}(f).
\]
For \(q\in\mathcal Q_{m,d}\), weighted arithmetic--geometric mean gives
\[
 (z_1+\cdots+z_m)^h
 \ge\frac{h^h z^{\one-q}}{(\one-q)^{\one-q}}.
\]
For \(h=0\) this is the identity \(1=1\).
Multiplying by \(f(z)\ge\Cap_q(f)z^q\) and applying
Corollary~\ref{cor:coefficient} to \(F\) proves the first inequality.

The capacity identity
\begin{equation}\label{eq:hypersimplex-capacity}
 \mathcal R(f)=\sup_{q\in\mathcal Q_{m,d}}\Cap_q(f)
\end{equation}
follows from \citet[Lemma~4.3]{AnariOveisGharanSaberiSingh2017} and uses only
nonnegative coefficients. To see it directly, we set \(z=e^u\) and minimize \(\log f(e^u)\)
subject to
\(\langle\one_S,u\rangle\ge0\) for every \(d\)-set \(S\).
This convex program has a strictly feasible point. If
\(\mathcal R(f)=0\), weak duality makes both sides of
\eqref{eq:hypersimplex-capacity} zero; otherwise its value is finite.
Strong duality identifies its value with the supremum of
\(\log\Cap_q(f)\), where
\(q=\sum_S\lambda_S\one_S\) and \(\lambda_S\ge0\).
Homogeneity forces \(\sum_S\lambda_S=1\), so these vectors fill
\(\mathcal Q_{m,d}\). Since
\((\one-q)^{\one-q}\le1\), this proves the second inequality.
Finally,
\[
 c_{m,d}=\prod_{r=h+1}^{m}
              \left(\frac{r-1}{r}\right)^{r-1}\ge e^{-d},
\]
where the factor for \(r=1\) is one.
\end{proof}

The averaging proof also records the slack in the coefficient step.
We view the identity matrix \(I_m\) as the deterministic word
\((1,\ldots,m)\) for \(F\). Since
\(\mathsf H_F(I_m)=\log(h!C_{\mathrm{sf}}(f))\), the equal-row identity
\eqref{eq:equal-entropy} and the definition \eqref{eq:Delta-def} give
\begin{equation*}
 C_{\mathrm{sf}}(f)=\frac{m!}{m^m h!}\Cap_{\one}(F)
                         \exp\bigl(\Delta_F(I_m)\bigr).
\end{equation*}
Thus the first two lower bounds in \eqref{eq:squarefree-sum} can
be multiplied by \(\exp(\Delta_F(I_m))\).
Theorem~\ref{thm:production} expresses this remainder through
nonnegative production along row averaging, with the boundary
interpreted by continuity. The numerical constant and the padding
argument are those of \citet{AnariOveisGharanSaberiSingh2017}.

For a product of linear forms, the matroid bounds of
Section~\ref{sec:matroids} improve the constant \(c_{m,d}\) to
\(d!/d^d\); the supplement derives this improvement and its consequence
for the allocation model of \citet{AnariOveisGharanSaberiSingh2017}
\citep[Section~S5.3]{sra2026supplement}.

\section{Binary averaging and stable pairings}
\label{sec:stable-pairings}

Row averaging compares optimized entropies at different marginals.
For a pairing of two polynomials, a second comparison keeps the
marginals fixed and averages a binary coupling toward independence.
Applied within the diagonal contractions of
\citet{AnariOveisGharan2017}, it proves their capacity bound with an
exact entropy remainder. Stability guarantees that successive
contractions satisfy the required two-variable inequality; for
permanents, matching switching supplies that inequality.

The coefficient pairing \(\sum_Sf_Sg_S\) arises from \(f(y)g(z)\) by
contracting the variables \(y_i\) and \(z_i\) in pairs, and the argument
tracks capacity through these contractions. The capacity definition and
the variational formula \eqref{eq:occupation-dual} apply also to
nonhomogeneous polynomials, and we write
\(h(t)=-t\log t-(1-t)\log(1-t)\) for the binary entropy. A real polynomial \(\Psi(y,z)\) in variables \(y,z\in\R^m\) is
\emph{bistable} if
\(\Psi(y,-z)\) is real stable; in this section \(\Psi\) has nonnegative
coefficients and is multiaffine in its \(2m\) variables. The diagonal
contraction
\[
\mathcal C_i\Psi=
\left.(1+\partial_{y_i}\partial_{z_i})\Psi\right|_{y_i=z_i=0}
\]
keeps the terms in which both selected variables occur or neither
occurs, and then deletes them.

\begin{lemma}[Diagonal contraction]\label{lem:diagonal-contraction}
If \(\Psi\) is bistable, then \(\mathcal C_i\Psi\) is bistable or zero.
For positive
 fixed values of the other variables, write the selected slice as
\[
 p(y_i,z_i)=a y_i z_i+b y_i+c z_i+d.
\]
Then \(ad\ge bc\).
\end{lemma}
\begin{proof}
For the preservation claim, we apply the sign change and write
\(R(u,v,w)=A(w)uv+B(w)u+C(w)v+D(w)\), which is stable.
Since \(-1/u\) lies in the upper half-plane whenever \(u\) does,
\[
 uR(u,-1/u,w)=B(w)u^2+(D(w)-A(w))u-C(w)
\]
is stable. Differentiating in \(u\) and specializing \(u=0\)
shows that \(D-A\) is stable or zero. Undoing the sign change
identifies this polynomial with \(\mathcal C_i\Psi\).
The preservation statement is \citet[Lemma~4.4]{AnariOveisGharan2017};
the argument here uses only differentiation, real specialization, and a
fractional linear substitution.

The bivariate polynomial \(p(y,-z)=-ayz+by-cz+d\) is stable or
zero by real specialization of the other variables. If \(ad<bc\),
then \(b,c>0\), and for \(y\) in the upper half-plane the root
\(z=(by+d)/(ay+c)\) also has positive imaginary part, since
\[
 \operatorname{Im}\frac{by+d}{ay+c}
 =\frac{bc-ad}{|ay+c|^2}\operatorname{Im}y>0.
\]
Such a root contradicts stability.
\end{proof}

\begin{lemma}[A binary entropy comparison]\label{lem:binary-pairing}
Let \(p(y,z)=ayz+by+cz+d\), where \(a,b,c,d\ge0\) and
\(ad\ge bc\). For \(0\le q\le1\),
\begin{equation}\label{eq:binary-capacity}
 a+d\ge e^{-h(q)}\Cap_{(q,q)}(p).
\end{equation}
If all four coefficients are positive and \(0<q<1\), let
\(\mu\) maximize weighted entropy with weights
\((w_{00},w_{10},w_{01},w_{11})=(d,b,c,a)\), subject to
\(\mathbb E X=\mathbb E Y=q\). Put
\(\lambda=\log(ad/(bc))\) and \(r=a/(a+d)\), and write
\(\operatorname{I}_\mu(X;Y)=\KL(\mu\Vert\mu_X\otimes\mu_Y)\). Then
\begin{align}\label{eq:binary-entropy-loss}
 \log(a+d)+h(q)-\log\Cap_{(q,q)}(p)
 &=\operatorname{I}_{\mu}(X;Y)
   +\lambda\mu(0,1)
   +\KL(\operatorname{Ber}(q)\Vert\operatorname{Ber}(r)).
\end{align}
Moreover, for
\[
 p_t(y,z)=ayz+e^{t\lambda/2}(by+cz)+d,
 \qquad 0\le t\le1,
\]
the capacity \(\Cap_{(q,q)}(p_t)\) is nondecreasing and
\begin{equation}\label{eq:binary-flow-loss}
 \log(a+d)+h(q)-\log\Cap_{(q,q)}(p)
 =\KL(\operatorname{Ber}(q)\Vert\operatorname{Ber}(r))
   +\int_0^1\lambda\mu_t(0,1)\,dt,
\end{equation}
where \(\mu_t\) is the corresponding weighted-entropy maximizer.
\end{lemma}
\begin{proof}
First suppose the four coefficients are positive and \(0<q<1\).
The constrained entropy maximizer has full support, and
\(\mu(0,1)=\mu(1,0)=s\). By \eqref{eq:occupation-dual},
\begin{align*}
 \log\Cap_{(q,q)}(p)
 &=\Ent(\mu)+(1-q)\log d+q\log a-s\lambda\\
 &=2h(q)-\operatorname{I}_{\mu}(X;Y)
       +(1-q)\log d+q\log a-s\lambda.
\end{align*}
Subtracting from \(\log(a+d)+h(q)\) gives
\eqref{eq:binary-entropy-loss}, since the remaining Bernoulli term
is relative entropy. All three terms are nonnegative.

For the interpolation, the coefficient ratio is
\(ad/(b_tc_t)=e^{(1-t)\lambda}\ge1\), so the slices remain
bistable. Strict concavity and the implicit function theorem give a unique
positive maximizer depending smoothly on \(t\). Differentiating the
variational optimum gives
\[
 \frac{d}{dt}\log\Cap_{(q,q)}(p_t)
 =\frac\lambda2\bigl(\mu_t(1,0)+\mu_t(0,1)\bigr)
 =\lambda\mu_t(0,1)\ge0.
\]
At \(t=1\), the coefficient array has rank one, so
\(X\) and \(Y\) are independent under its Gibbs law and
\[
 \log\Cap_{(q,q)}(p_1)
 =2h(q)+(1-q)\log d+q\log a.
\]
Integration proves \eqref{eq:binary-flow-loss}.

For the remaining cases, \(q=0,1\) gives capacities \(d,a\),
respectively. If \(ad>0\) but \(bc=0\), any supported law with
 equal marginals is diagonal, so
\(\log\Cap_{(q,q)}(p)=h(q)+(1-q)\log d+q\log a\);
weighted arithmetic--geometric mean gives even
\(\Cap_{(q,q)}(p)\le a+d\). If \(ad=0\), then \(bc=0\)
and no supported law has equal marginals in \((0,1)\), so that
capacity is zero. These cases prove \eqref{eq:binary-capacity}
in full generality.
\end{proof}

The interpolation averages the binary coupling toward independence.
Indeed, if \(s_t=\mu_t(0,1)\), its Gibbs odds ratio is
\[
 \frac{(q-s_t)(1-q-s_t)}{s_t^2}=e^{(1-t)\lambda}.
\]
The left side strictly decreases with \(s_t\), so \(s_t\) increases
from \(s_0\) to \(q(1-q)\). For \(\lambda>0\), this gives
\[
 \mu_t=(1-\theta_t)\mu_0+\theta_t
       \bigl(\operatorname{Ber}(q)\otimes\operatorname{Ber}(q)\bigr),
 \qquad
 \theta_t=\frac{s_t-s_0}{q(1-q)-s_0}\in[0,1].
\]
When \(\lambda=0\), the coupling is already independent and the
path is constant. The averaging comparison is local to each bivariate
slice; it does not require a simultaneous interpolation of all
coefficients of \(\Psi\). Iteration requires the condition \(ad\ge bc\)
to persist under contraction.

Iterating Lemma~\ref{lem:binary-pairing} along the diagonal contractions
gives the multiaffine pairing bound of
\citet[Theorem~1.1]{AnariOveisGharan2017}, now with its exact loss.
Lemma~\ref{lem:diagonal-contraction} supplies the closure that the
iteration needs.

\begin{theorem}[Multiaffine stable pairings]\label{thm:multiaffine-pairing}
Let \(f,g\) be multiaffine real stable polynomials in \(m\) variables
with nonnegative coefficients, and let \(q\in[0,1]^m\). Then
\begin{equation}\label{eq:ao-multiaffine}
 \sum_{S\subseteq[m]}f_Sg_S
 \ge q^q(\one-q)^{\one-q}\Cap_q(f)\Cap_q(g).
\end{equation}
Suppose both capacities are positive. Put \(\Phi(y,z)=f(y)g(z)\) and
\(\Phi_i=\mathcal C_i\cdots\mathcal C_1\Phi\), and write
\(\Cap(\Phi_i)\) for the capacity of \(\Phi_i\) at the exponents
\(q_j\) of \(y_j\) and \(z_j\), \(j>i\). Then the slacks
\[
 \delta_i=\log\Cap(\Phi_i)-\log\Cap(\Phi_{i-1})+h(q_i)
\]
are nonnegative, and
\begin{equation}\label{eq:pairing-exact-loss}
 \sum_{S\subseteq[m]}f_Sg_S
 =q^q(\one-q)^{\one-q}\Cap_q(f)\Cap_q(g)
       \exp\Bigl(\sum_{i=1}^m\delta_i\Bigr).
\end{equation}
\end{theorem}
\begin{proof}
The polynomial \(\Phi\) is bistable, because real coefficients imply
that a stable polynomial is also nonvanishing when all variables lie in
the lower half-plane. For a bistable \(\Psi\), we fix all variables
other than \(y_i,z_i\) at positive values. By
Lemma~\ref{lem:diagonal-contraction} the resulting slice satisfies
\(ad\ge bc\), so \eqref{eq:binary-capacity} applies to it. Taking the
infimum over the fixed variables gives
\begin{equation}\label{eq:contraction-capacity}
 \Cap_{(q_{-i},q_{-i})}(\mathcal C_i\Psi)
 \ge e^{-h(q_i)}\Cap_{(q,q)}(\Psi),
\end{equation}
where \(q_{-i}\) omits the \(i\)th coordinate. Every \(\Phi_i\) is
bistable or zero by Lemma~\ref{lem:diagonal-contraction}, so
\eqref{eq:contraction-capacity} applies at each step. If both
capacities of \(f\) and \(g\) are positive, then inductively every
\(\Cap(\Phi_i)\) is positive and \(\delta_i\ge0\). The full contraction
is
\[
 \Phi_m=\sum_{S\subseteq[m]}f_Sg_S,
 \qquad
 \Cap(\Phi_0)=\Cap_q(f)\Cap_q(g),
\]
so the slacks telescope to \eqref{eq:pairing-exact-loss}. Since
\(e^{-\sum_ih(q_i)}=q^q(\one-q)^{\one-q}\), nonnegativity of the
slacks gives \eqref{eq:ao-multiaffine}, which is immediate when one
capacity is zero.
\end{proof}

Each slack separates into a local binary loss and a minimization loss.
Let \(v^{(k)}\) be a minimizing sequence for \(\Cap(\Phi_i)\), and put
\(K_i(v)=\Cap_{(q_i,q_i)}(\Phi_{i-1}(v,\cdot,\cdot))\).
Writing \(v=(y^{\mathrm{rem}},z^{\mathrm{rem}})\) for the variables with
index \(j>i\) and
\(v^{q^{\mathrm{rem}}}=(y^{\mathrm{rem}}z^{\mathrm{rem}})^{q_{i+1:m}}\)
for the remaining denominator,
\[
 \delta_i=\lim_{k\to\infty}\left[
  \log\frac{\Phi_i(v^{(k)})e^{h(q_i)}}{K_i(v^{(k)})}
 +\log\frac{K_i(v^{(k)})}
                 {(v^{(k)})^{q^{\mathrm{rem}}}\Cap(\Phi_{i-1})}\right].
\]
Both terms are nonnegative. For positive local coefficients the
first has the representations
\eqref{eq:binary-entropy-loss}--\eqref{eq:binary-flow-loss}.
Thus no attainment assumption is needed for the capacity minima.
The total loss is independent of the elimination order, although
its individual terms need not be.

By algebraic polarization, \citet[Theorem~4.1]{AnariOveisGharan2017}
extend \eqref{eq:ao-multiaffine} to all real stable polynomials with
nonnegative coefficients, with the factor \(e^{-|q|}q^q\) for the
pairing \(\sum_\alpha\alpha!f_\alpha g_\alpha\); the supplement derives
this extension from Theorem~\ref{thm:multiaffine-pairing}
\citep[Section~S5.1]{sra2026supplement}.

\subsection{Retaining common linear constraints}
\label{sec:pairing-constraints}

A contraction need not lose one binary-entropy factor at every
coordinate. A common linear constraint can force the last variables
to agree, so they can be summed without that loss. Retaining such
coordinates gives an explicit correction, with no intermediate
capacities to compute.

\begin{proposition}[Common linear constraints]\label{prop:pairing-constraints}
Under the assumptions of Theorem~\ref{thm:multiaffine-pairing}, suppose
that a real matrix \(W\) and a vector \(c\) satisfy
\[
 W\one_A=c\quad\text{whenever }f_A>0\text{ or }g_A>0.
\]
If the columns of \(W\) indexed by \(S\subseteq[m]\) are
linearly independent, then, for every \(q\in[0,1]^m\),
\begin{equation}\label{eq:pairing-constraints}
 \sum_A f_Ag_A
 \ge\exp\left(-\sum_{j\notin S}h(q_j)\right)
          \Cap_q(f)\Cap_q(g).
\end{equation}
In particular, when \(f,g\) are homogeneous of the same degree,
\begin{equation}\label{eq:homogeneous-pairing-correction}
 \sum_A f_Ag_A
 \ge q^q(\one-q)^{\one-q}
       \Cap_q(f)\Cap_q(g)\exp\left(\max_j h(q_j)\right).
\end{equation}
\end{proposition}

\begin{proof}
If either capacity is zero, the claim is immediate. Otherwise we start
with \(\Phi(y,z)=f(y)g(z)\) and contract every coordinate outside \(S\). A surviving term originated from a pair \(A,B\)
that agrees outside \(S\). The common constraints give
\(W_S(\one_{A\cap S}-\one_{B\cap S})=0\), so independence
of the selected columns forces \(A\cap S=B\cap S\).
Thus the remaining polynomial \(Q\) contains only terms
\(y^Tz^T\), and its full contraction is \(Q(\one,\one)\).
Evaluation at \((\one,\one)\) gives
\(Q(\one,\one)\ge\Cap_{(q_S,q_S)}(Q)\).
Applying \eqref{eq:contraction-capacity} to the coordinates outside
\(S\) gives \eqref{eq:pairing-constraints}. For homogeneous polynomials
we take \(W=(1,\ldots,1)\) and leave uncontracted one coordinate that
maximizes \(h(q_j)\).
\end{proof}

If both polynomials have the same degree in each block of a partition,
we may retain one coordinate from each block. The correction in
\eqref{eq:homogeneous-pairing-correction} then becomes the sum of the
largest binary entropy in each block.

\subsection{The Bethe bound and Schrijver's inequality}
\label{sec:schrijver-contraction}

We represent a permanent as a pairing by giving each edge one variable
in a row selector and one in a column selector. The common monomials are
then exactly the perfect matchings, and
\citet[Section~3.1]{AnariOveisGharan2017} derive Schrijver's inequality
from their pairing bound in this way. The Bethe lower bound is stated
through the following functional: for \(B>0\) and a doubly stochastic
\(P\), we write
\[
 H_{\mathrm B}(P)=\sum_{i,j}
 \bigl[-p_{ij}\log p_{ij}+(1-p_{ij})\log(1-p_{ij})\bigr],
 \qquad
 \beta_B(P)=\langle P,\log B\rangle+H_{\mathrm B}(P).
\]
The entropy conventions extend these expressions to zero entries
whenever \(\supp P\subseteq\supp B\).

For the permanent, a switching argument supplies every local inequality
needed by the binary-entropy comparison, without stability. We record
this elementary form
of the monomer--dimer correlation identity of Heilmann and Lieb
\citep{HeilmannLieb1972}.

\begin{lemma}[Matching switching]\label{lem:matching-switching}
Let \(G=(R\sqcup C,F)\) be bipartite, with nonnegative edge weights
\(w_e\) and vertex weights \(\xi_v\), and put
\[
 Z_G=\sum_{M\text{ matching in }G}
       \prod_{e\in M}w_e\prod_{v\notin V(M)}\xi_v.
\]
If \(r\in R\) and \(c\in C\), then
\begin{equation}\label{eq:matching-switching}
 Z_G Z_{G-r-c}\ge Z_{G-r}Z_{G-c}.
\end{equation}
All deleted graphs inherit the remaining vertex and edge weights,
and \(Z_{\varnothing}=1\). More precisely, the defect is a sum over the
simple paths \(\gamma\) from \(r\) to \(c\),
\begin{equation}\label{eq:matching-path-defect}
 Z_G Z_{G-r-c}-Z_{G-r}Z_{G-c}
 =\sum_{\gamma:r\leadsto c}
       \left(\prod_{e\in\gamma}w_e\right)Z_{G-V(\gamma)}^2.
\end{equation}
\end{lemma}

\begin{proof}
Color a matching of \(G-r\) red and a matching of \(G-c\) blue.
Their union consists of alternating paths and cycles, with a doubled
edge regarded as a cycle. The component containing \(r\) cannot
contain \(c\), because a path joining them would start blue and end red,
so would have even length, whereas every such path in \(G\) has
odd length. Interchange the colors on this component. The result
is a red matching of \(G\) and a blue matching of \(G-r-c\).
This map preserves the product of the edge and vertex weights, and it
is injective because the same interchange recovers its input.
When \(r\) is isolated in the union, its unmatched-vertex weight
simply moves from the blue matching to the red matching.
Summing the preserved weights proves \eqref{eq:matching-switching}.

The pairs outside the image have an alternating path from \(r\)
to \(c\), starting and ending red. This path is unique and has
odd length. For a fixed path \(\gamma\), its colors are forced;
deleting it leaves two arbitrary matchings of \(G-V(\gamma)\).
Their total weight is the corresponding summand in
\eqref{eq:matching-path-defect}, which proves the identity.
\end{proof}

\begin{corollary}[A Bethe bound with a binary-entropy correction]
\label{cor:bethe-correction}
Let \(B\ge0\) and let \(P\) be doubly stochastic with
\(\supp P\subseteq\supp B\). If \(E_1,\ldots,E_s\) are
the edge sets of the connected components of the bipartite support
graph of \(B\), then
\begin{equation}\label{eq:bethe-correction}
 \log\per B\ge\beta_B(P)
               +\sum_{a=1}^s\max_{e\in E_a}h(p_e).
\end{equation}
In particular, dropping the correction recovers the Bethe lower
bound. For every doubly stochastic \(A\),
\begin{equation}\label{eq:schrijver}
 \per\bigl(A\odot(\one\one^T-A)\bigr)
 \ge\prod_{i,j}(1-a_{ij}).
\end{equation}
\end{corollary}

\begin{proof}
We follow the contraction argument of
Theorem~\ref{thm:multiaffine-pairing}, with matching switching in place
of stability. We use variables indexed by the positive entries of
\(B\), and put
\[
 f(y)=\prod_i\left(\sum_{j:b_{ij}>0}b_{ij}y_{ij}\right),
 \qquad
 g(z)=\prod_j\left(\sum_{i:b_{ij}>0}z_{ij}\right).
\]
Their coefficient pairing equals \(\per B\). To apply the
binary-entropy comparison without a stability assumption, we contract
an arbitrary set \(F\) of edge-variable pairs and write
\(G_F=(R\sqcup C,F)\). Expansion of the row and column selectors
gives the exact invariant
\begin{equation*}
 \left(\prod_{e\in F}\mathcal C_e\right)f(y)g(z)
 =Z_{G_F}(\xi),\qquad
 \xi_{r_i}=\sum_{j:(i,j)\notin F}b_{ij}y_{ij},\quad
 \xi_{c_j}=\sum_{i:(i,j)\notin F}z_{ij}.
\end{equation*}
Here \(Z_{G_F}\) carries the edge weights \(b_e\) and the vertex
weights \(\xi\), and the sums use only the support of \(B\). Indeed, a
contracted edge
must be selected at both ends or at neither end. The edges selected
at both ends form a matching, and each unmatched vertex selects one
of its remaining half-edges.

For an uncontracted edge \(e=(r,c)\), we fix all other variables
positively and write \(\xi_r=u+b_e y_e\),
\(\xi_c=v+z_e\). The resulting slice is
\(ay_ez_e+by_e+cz_e+d\), with
\[
 a=b_eZ_{G_F-r-c},\qquad b=b_eZ_{G_F-r},\qquad
 c=Z_{G_F-c},\qquad d=Z_{G_F},
\]
where the vertex weights at \(r,c\) in these four partition
functions are \(u,v\). Lemma~\ref{lem:matching-switching}
therefore gives \(ad\ge bc\). We apply Lemma~\ref{lem:binary-pairing} at
each contraction and take infima over the remaining variables, exactly
as in \eqref{eq:contraction-capacity}. The terminal common-constraint
argument of Proposition~\ref{prop:pairing-constraints} uses only
the surviving monomials, so it also applies here. Thus the
component correction follows from binary entropy and matching
switching alone.

Since each variable occurs in only one factor of either polynomial,
weighted arithmetic--geometric mean gives
\[
 \Cap_P(f)=\prod_{i,j}\left(\frac{b_{ij}}{p_{ij}}\right)^{p_{ij}},
 \qquad
 \Cap_P(g)=\prod_{i,j}p_{ij}^{-p_{ij}}.
\]
Equality is attained or approached at the corresponding row or column
proportions. Products are over the support of \(B\), and zero exponents
are interpreted by continuity. Every support component has equally many
row and column vertices, because \(P\) has unit row and column
sums. Thus \(f,g\) have the same degree in each component's
variables. In the preceding contraction argument, we retain an edge of
largest binary entropy in each component. The logarithm of
the uncorrected right side is
\[
 \sum_{i,j}\bigl[p_{ij}\log b_{ij}-p_{ij}\log p_{ij}
                   +(1-p_{ij})\log(1-p_{ij})\bigr]
 =\beta_B(P),
\]
which proves \eqref{eq:bethe-correction}.

For \(0<a_{ij}<1\), we take \(P=A\) and \(b_{ij}=a_{ij}(1-a_{ij})\), and
drop the nonnegative correction. The exponent simplifies to
\(\beta_B(A)=\sum_{i,j}\log(1-a_{ij})\), proving
\eqref{eq:schrijver}. Approximation by
\((1-\varepsilon)A+\varepsilon\one\one^T/n\) and continuity
handle the boundary; the case \(n=1\) is immediate.
\end{proof}

Equation~\eqref{eq:schrijver} is Schrijver's inequality
\citep{Schrijver1998}. Its variational form without the correction is
due to Gurvits \citep[Theorem~2.2]{GurvitsSchrijver2012}, and
Anari and Oveis Gharan derive it from their pairing bound. Here the
switching lemma replaces their stability input, and
Lemma~\ref{lem:binary-pairing} retains the entropy loss that gives the
component correction. Through the same multiaffine inequality of Anari and Oveis Gharan,
\citet[Theorem~6]{Anari2026} multiplies the Bethe lower bound by gains
from pairs of rows and obtains a deterministic
approximation of the permanent within a factor \(c^n\) for some
\(c<\sqrt2\) \citep[Theorem~1]{Anari2026}. Row averaging does not make
\(\mathsf H_{\mathrm{perm}}-H_{\mathrm B}\) monotone
(Appendix~\ref{chain:bethe-gap}), so the correction comes from the
binary comparison rather than from Theorem~\ref{chain:monotonicity}.
The switching inequality also enters the entropy interpolation of
\citet[Claim~2.5 and Theorem~2.1]{DongJain2026}, whose parameter
is the probability that a vertex is covered by a matching.

The path identity locates the local interaction, since it gives
\[
 ad-bc=b_e\sum_{\gamma:r\leadsto c\text{ in }G_F}
       \left(\prod_{f\in\gamma}b_f\right)
       Z_{G_F-V(\gamma)}^2.
\]
The local determinant is therefore positive only if the contracted
edge closes a cycle with previously contracted edges, and with
positive vertex weights every such path contributes strictly. These
cycle terms enter the entropy-production formula
\eqref{eq:binary-flow-loss} through \(\lambda=\log(ad/(bc))\).

The same correction applies directly to the maximum permutation
entropy \(\mathsf H_{\mathrm{perm}}\). Let \(E_1,\ldots,E_s\) now be the support components of a
doubly stochastic \(P\). Finite Gibbs duality on
this face gives
\[
 \mathsf H_{\mathrm{perm}}(P)
 =\inf_{\substack{b_{ij}>0\text{ on }\supp P\\b_{ij}=0\text{ elsewhere}}}
       \{\log\per B-\langle P,\log B\rangle\}.
\]
Indeed, the same maximum-entropy argument as in
Lemma~\ref{chain:gibbs} applies after restricting to the allowed
permutations and removing their affine relations. Taking this infimum
in \eqref{eq:bethe-correction} yields
\begin{equation}\label{eq:bethe-entropy-correction}
 \mathsf H_{\mathrm{perm}}(P)
 \ge H_{\mathrm B}(P)+\sum_{a=1}^s\max_{e\in E_a}h(p_e).
\end{equation}
This bound is exact whenever every support component is a single
edge or an even cycle. A cycle has two perfect matchings; its
marginals alternate between \(t\) and \(1-t\), so its maximum
entropy is \(h(t)\), its Bethe entropy is zero, and its correction
is \(h(t)\). Distinct components can be made independent, which
adds their maximum entropies. A single edge contributes zero.

For example, let \(B\) be the adjacency matrix of an \(r\)-regular
bipartite graph on \(n+n\) vertices, with \(s\) connected
components and \(r\ge2\). Taking \(P=B/r\) gives
\[
 \per B\ge e^{s h(1/r)}
       \left(\frac{(r-1)^{r-1}}{r^{r-2}}\right)^n.
\]
When \(r=2\), every component is an even cycle, and the bound \(2^s\)
is exact.

\section{Further directions}\label{sec:directions}

\paragraph{Effective remainders.}
The explicit quadratic corrections of Section~\ref{sec:geometry} keep
the slack of the partial bounds at order two. A different route is to
bound the negative kernel eigenvalues in
Corollary~\ref{cor:spectral-remainder} directly from the input data,
which could give effective entropy remainders even when the minimum
entry or eigenvalue vanishes.

\paragraph{Degree-sensitive coefficient bounds.}

Let \(p\) be a stable homogeneous polynomial of degree \(n\) in \(n\)
variables with nonnegative coefficients and positive capacity. Gurvits's
degree-sensitive bound is
\[
 [x_1\cdots x_n]p\ge\Cap_{\one}(p)
       \prod_{i=1}^n g\bigl(\min\{i,\deg_{x_i}p\}\bigr),
 \qquad
 g(r)=\left(\frac{r-1}{r}\right)^{r-1},\quad g(1)=1,
\]
with an arbitrary ordering of the variables \citep{Gurvits2008}.
This refinement does not hold for all Lorentzian polynomials.

To see this, we set \(s=y+z\) and consider
\[
 f(x,y,z)=s^3+3xs^2+3x^2s.
\]
Its support is
\(\{\alpha\in\mathbb Z_{\ge0}^3:|\alpha|=3,\ \alpha_x\le2\}\),
an \(M\)-convex set. Moreover,
\[
 \partial_xf=3s^2+6xs,\qquad
 \partial_yf=\partial_zf=3(x+s)^2.
\]
The first Hessian has one positive, one negative, and one zero
eigenvalue, and the other two have rank one and are positive
semidefinite. Thus \(f\) is Lorentzian. Its mixed coefficient is
\([xyz]f=6\), whereas \(yz\le s^2/4\) gives
\[
 \Cap_{(1,1,1)}(f)
 =4\inf_{t>0}\bigl(t^{-1}+3+3t\bigr)
 =4(3+2\sqrt3).
\]
Equality in the first reduction is attained at \(y=z\), and the
one-variable minimum is attained at \(t=x/s=1/\sqrt3\).
Ordering the variables as \(y,z,x\) gives degree factors
\(g(1)g(2)g(2)=1/4\), which would require
\(6\ge3+2\sqrt3\). An averaging proof of the degree-sensitive
bound must therefore use additional structure of stable polynomials.

\paragraph{Pairings beyond bistability.}

The diagonal contractions in Section~\ref{sec:stable-pairings} supply
the additional stable-polynomial input needed for the Anari--Oveis
Gharan pairing inequalities. Corollary~\ref{cor:refined-lower} applies
to a different class, a Lorentzian factor paired with a product of
linear forms. Extending its prescribed-marginal comparison to two
general Lorentzian factors requires a replacement for bistable
contraction; the coefficient-pairing problem is discussed by
\citet[Conjecture~6.1]{GurvitsLeake2021}.

The direct coefficientwise product does not preserve the quadratic
condition required by row averaging. Indeed, both
\[
 f=(x_1+x_2)(x_3+x_4),\qquad
 g=(x_1+x_3)(x_2+x_4)
\]
are stable, but the product of their corresponding coefficients is
\[
 \sum_\alpha [x^\alpha]f\,[x^\alpha]g\,x^\alpha
 =x_1x_4+x_2x_3.
\]
Its Hessian has two positive eigenvalues. Retaining separate variables
for the two factors avoids this product in the stable case. In the
matching specialization, switching supplies the required local
inequality without stability, which suggests seeking correlation
inequalities for other supports that allow binary averaging to be
iterated.

\paragraph{Real linear polarization.}
The factorial constant of Corollary~\ref{cor:coefficient} does not give
the real polarization inequality for products of linear functionals on
a sphere, because signs enter as well. The supplement isolates this
obstruction and proves a variational special case
\citep[Section~S6.3]{sra2026supplement}.

\begin{aiusage}
The results in this paper were developed through extensive use of an AI
assistant (OpenAI's GPT~Pro, in a session that identified itself as
GPT-6 Astra Pro). None of the claims were obtained zero-shot. The
author's proposal of a Schur-type averaging statement for permanents led
to Theorem~\ref{chain:monotonicity}; at the author's request the
argument was compared with the proofs of Falikman, Egorychev, and
Gurvits, and tested on mixed volumes and Lorentzian polynomials, leading
to Sections~\ref{sec:lorentzian}--\ref{sec:applications}. The
AI assistant carried out the derivations, the literature search, and
checks of the arguments. The partial-matching and binary-averaging
results (Sections~\ref{sec:partial-matchings}
and~\ref{sec:stable-pairings}) were developed later in the same way, at
the author's request, with AI assistants carrying out the derivations
and independent checks. The author also used AI assistants
(GPT and Claude) to polish the writing.
\end{aiusage}

\aiacknowledgements

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\appendix
\section{Counterexamples}\label{chain:scope}

Theorem~\ref{chain:monotonicity} is stated for chains of pair averages of
entire rows or columns. The examples below show that this restriction is
needed, and that the Bethe entropy gap of
Section~\ref{sec:schrijver-contraction} is not monotone even under such
averages.

\subsection{Majorization of the entries}

The order generated by pair averages is not majorization of the
flattened list of entries. For example, let \(S\) be a cyclic permutation
matrix of order four. The doubly stochastic matrices
\[
 A=\diag(J_2/2,J_2/2),\qquad B=(I_4+S)/2
\]
both have eight entries equal to \(1/2\) and eight zero
entries, while their maximum-entropy permutation laws are uniform on four
and two supported permutations, respectively. Consequently
\begin{equation*}
 \mathcal I(A)=2\log2,\qquad \mathcal I(B)=3\log2,
\end{equation*}
so \(\mathcal I\) is not a symmetric function of the flattened
entries, as ordinary Schur-concavity would require.

\subsection{Concavity and arbitrary stochastic mixing fail}

Nor can the pair-averaging theorem be replaced by a concavity assertion.
Let \(B(t)\) be the \(3\times3\) matrix with diagonal \(t\) and
off-diagonal entries \(1\). Its Gibbs marginal matrix \(P(x)\) has
diagonal
\[
 x=\frac{t(t^2+1)}{t^3+3t+2}
\]
and off-diagonal entries \((1-x)/2\). If \(N\) is the number of
fixed points under this Gibbs law, differentiation of the exponential
family gives
\[
 \frac{d^2}{dx^2}\mathcal I(P(x))
 =\frac9{\operatorname{Var}N}-\frac3{x(1-x)},
\]
because \(\mathbb E N=3x\),
\(d(3x)/d\log t=\operatorname{Var}N\), and
\(d\mathsf H_{\mathrm{perm}}(P(x))/dx=-3\log t\).
At \(t=1/3\),
\begin{equation*}
 x=\frac5{41},\qquad
 \operatorname{Var}N=\frac{513}{1681},\qquad
 \frac{d^2}{dx^2}\mathcal I(P(x))=\frac{1681}{1140}>0.
\end{equation*}

Arbitrary doubly stochastic left multiplication cannot replace a chain
of pair \(T\)-transforms either. Fix \(r=3/4\), and put
\[
 B_\varepsilon=
 \begin{pmatrix}\varepsilon&1&1\\1&\varepsilon&1\\r&1&\varepsilon\end{pmatrix},
 \qquad
 Z_\varepsilon=1+r+(2+r)\varepsilon+\varepsilon^3.
\]
Its cofactor matrix is
\[
 C_\varepsilon=
 \begin{pmatrix}
 1+\varepsilon^2&r+\varepsilon&1+r\varepsilon\\
 1+\varepsilon&r+\varepsilon^2&r+\varepsilon\\
 1+\varepsilon&1+\varepsilon&1+\varepsilon^2
 \end{pmatrix},
\]
so \(P_\varepsilon=(B_\varepsilon\odot C_\varepsilon)/Z_\varepsilon\)
is an interior doubly stochastic matrix. Let \(\Pi\) cyclically
reorder its rows as \((2,3,1)\). Formula
\eqref{chain:differential} gives
\[
 d\mathcal I(P_\varepsilon)[\Pi P_\varepsilon-P_\varepsilon]
 =-\sum_{i,j}(\Pi P_\varepsilon-P_\varepsilon)_{ij}
                  \log(C_\varepsilon)_{ij}.
\]
As \(\varepsilon\downarrow0\),
\[
 P_\varepsilon\longrightarrow\frac1{1+r}
 \begin{pmatrix}0&r&1\\1&0&r\\r&1&0\end{pmatrix},
\]
and the limiting matrix \(-\log C_\varepsilon\) is zero except at
\((1,2),(2,2),(2,3)\), where it equals \(-\log r\). Therefore
\begin{equation*}
 \lim_{\varepsilon\downarrow0}
 d\mathcal I(P_\varepsilon)[\Pi P_\varepsilon-P_\varepsilon]
 =(-\log r)\frac{1-2r}{1+r}
 =-\frac27\log\frac43<0.
\end{equation*}
Hence the doubly stochastic matrix \((1-t)I+t\Pi\) decreases
\(\mathcal I\) for small \(\varepsilon>0\) and then small \(t>0\), so the
pair-averaging order in Theorem~\ref{chain:monotonicity} is a genuine
restriction.

\subsection{The Bethe entropy gap is not monotone}\label{chain:bethe-gap}

Let \(\mathcal J(P)=\mathsf H_{\mathrm{perm}}(P)-H_{\mathrm B}(P)\),
with \(H_{\mathrm B}\) as in Section~\ref{sec:schrijver-contraction}.
Equation~\eqref{eq:bethe-entropy-correction} bounds this gap by diagonal
contractions. Row averaging need not increase this gap, even in the
interior of the doubly stochastic polytope.

Consider the Gibbs matrix \(B\), with \(\per B=377\), and its marginals \(P\):
\[
 B=\begin{pmatrix}
 3&3&1&2\\1&1&2&3\\1&1&3&3\\2&3&1&1
 \end{pmatrix},\qquad
 P=\frac1{377}\begin{pmatrix}
 168&123&32&54\\48&39&122&168\\
 39&32&183&123\\122&183&40&32
 \end{pmatrix}.
\]
Let \(u,v\) be its first two rows and \(x,y\) the first two
rows of \(B\). Along the path replacing \(u,v\) by
\((1-t)u+tv,\ tu+(1-t)v\), the entropy differential gives
\[
 \left.\frac{d}{dt}\mathcal J(P(t))\right|_{t=0}
 =\sum_j(v_j-u_j)
       \log\frac{(1-u_j)y_ju_j}{(1-v_j)x_jv_j}.
\]
Substitution yields the exact expression
\[
 377\left.\frac{d}{dt}\mathcal J(P(t))\right|_{t=0}
 =6\log R,\qquad
 R=\left(\frac{282}{209}\right)^{20}
   \left(\frac{6591}{5207}\right)^{14}
   \left(\frac{736}{1037}\right)^{15}
   \left(\frac{459}{616}\right)^{19}<\frac14.
\]
The final inequality follows by clearing the positive denominators.
Thus \(\mathcal J\) decreases under a sufficiently small pair
average. This counterexample concerns the proposed monotonicity;
the Bethe lower bound remains valid.

\end{document}