The note's source

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\hypnostitle{Maximal Coherence for Prescribed Intrinsic Populations and Youla Values: Two Questions of Gil}
\hypnosshorttitle{Maximal coherence for prescribed populations and Youla values}
\hypnoswriter{Claude Fable 5.1 (Anthropic)}
\hypnosdate{September 29, 2026}

\begin{document}
\hypnosmaketitle
\begin{abstract}
In the intrinsic reference basis of a finite-dimensional density matrix,
$\rho=A+iN$ with $A=\diag(a_1\ge\dots\ge a_n\ge0)$ the intrinsic populations
and $N$ real antisymmetric; Gil (Entropy 28:877, 2026) calls $\|N\|_\F^2$ the
squared cohesion. Gil showed that among states whose metaspin tensor is
supported on a matching of intrinsic axes (the aligned class) the squared
cohesion is maximized, for fixed populations, by pairing adjacent populations,
and asked whether this remains true over all orientations of the Youla frame
with free Youla values, and what the maximum over all orientations is when
the Youla values are prescribed. We determine both maxima. With free Youla
values the maximum over all states with these populations is
$2\sum_ka_{2k-1}a_{2k}$, so Gil's adjacent-pairing value is the global
maximum. With prescribed Youla values $1\ge s_1\ge\dots\ge s_m\ge0$,
$m=\lfloor n/2\rfloor$, the maximum over all orientations is
$2\sum_ks_k^2a_{2k-1}a_{2k}$, attained by placing the $k$-th largest Youla
value on the $k$-th adjacent pair; when populations vanish, both statements
are read on their support. The second value rests on the inequality
$\sum_{i<j}a_ia_j|M_{ij}|^2\le\sum_ks_k^2a_{2k-1}a_{2k}$ for every
skew-symmetric $M$ with singular values $s_1,s_1,\dots,s_m,s_m$, which
follows from Horn's 1950 inequality for the singular values of a product and
is the Frobenius case of Mathias's 1992 inequality for Hadamard products; we
claim no priority for it. Two proofs are given. For free values, the squared
entries of a real antisymmetric contraction lie in Edmonds' matching
polytope, which extends the statement to arbitrary nonnegative pairwise
weights; for prescribed values, we use a layer-cake decomposition with Ky
Fan's maximum principle and interlacing of singular values, which yields a family
of linear inequalities on the squared entries of skew matrices with
prescribed spectrum. The product form of the weights is essential there:
with prescribed Youla values and general nonnegative weights the maximum need
not be aligned, as an explicit example with $n=4$ shows. Gil's question on
the image of the population-weighted Gram map, and the feasibility problem
with the spectrum of $\rho$ fixed, are not addressed.
\end{abstract}

\begin{attribution}
This paper was written by Claude Fable 5.1, an AI model made by Anthropic, at the direction of David Ross. The model found the two questions of Gil while reviewing the companion manuscript on approximate antiunitary symmetry (whose hull lemma it reuses), proved the theorems, wrote the verification program and the text; David Ross set the task, ran the process, and takes responsibility for the manuscript. The answer to the free-value question (Theorem~\ref{thm:free}, eq.~\eqref{eq:free}) was also found and proved independently, through the same hull lemma, by the companion manuscript's adversarial self-review (GPT-6 Astra, 2026-09-28), before this note's first draft; neither review had read the other, and the companion's Section~1 records that both found the application. Four reviews were applied, each with a per-item ledger. On 2026-09-29, Claude Opus 5.5 (Anthropic), in a fresh context, found the proofs complete and three framing defects (an overclaimed abstract, refuted by the $n=4$ example it supplied, now Remark~\ref{rem:nonproduct}; an unread source behind a novelty claim; a misreported numerical search); GPT-6 Astra (OpenAI), through the Codex CLI, found no blocking defect and two major items (Theorem~\ref{thm:prescribed} follows from Horn's inequality, now its second proof; Corollary~\ref{cor:consequences}(1) needed its support restriction) and supplied the exact certificates of Remark~\ref{rem:nonproduct}. A consistency pass on 2026-10-01 by Claude Opus 5.5 checked the exposition and the quotations against their sources. On October 2, GPT-6 Astra, fresh to the note, found both theorems and the hull lemma correct and corrected the exact-orbit certificate of Remark~\ref{rem:nonproduct} (a factor of two and the sign of the Pfaffian; the stated maximum unchanged) and a quotation locator; Claude Opus 5.5, fresh to the note, found every numbered statement correct and two major items of attribution and source access (the earlier independent proof of Theorem~\ref{thm:free}; the disclosure about Mathias's abstract) and ten minor ones. All were applied. A literature pass the same day found Mathias's 1992 theorem, which contains inequality~\eqref{eq:prescribed} as its Frobenius case; Remark~\ref{rem:lit} cites it through its zbMATH review. The reviews and the per-item dispositions accompany the source. David Ross read every page, for the only things he can judge: that the account of the harness and of the process matches the record, and that nothing reads like a machine grading its own homework. He did not check the proofs, and could not have; a reader should take his reading as a check for red flags, not as a review. No human mathematician has reviewed this paper. It is one of three manuscripts that came from the record of Hypnos, the research harness described in Section~\ref{sec:provenance}, on the night of 2026-09-28/29, this one at one remove (the harness contributed no step of its mathematics); it is the last of the three, after Astra's antiunitary paper and the Riesz-basis paper on the unfolded zeta zeros, and it was written without reading the latter.
\end{attribution}

\section{The question}

Gil~\cite{gil} organizes a finite-dimensional density matrix $\rho$ by
diagonalizing its real part with a real orthogonal congruence. In the
resulting intrinsic reference basis (IRB),
\begin{equation}\label{eq:irb}
\rho=A+iN,\qquad A=\diag(a_1,\dots,a_n),\quad a_1\ge\dots\ge a_n\ge0,\quad \sum_ia_i=1,\quad N^T=-N,
\end{equation}
with $N$ real (Gil, eq.~(3)); a density matrix is Hermitian, positive
semidefinite and trace-normalized (Gil, Section 2.1), and the IRB is obtained
from a fixed real structure by a real orthogonal change of basis, so the
$a_i$, the intrinsic populations, are the eigenvalues of the real part of
$\rho$ and not in general those of $\rho$. For strictly positive populations
the metaspin tensor is
\begin{equation}\label{eq:metaspin}
M_{ij}=\frac{N_{ij}}{\sqrt{a_ia_j}}\qquad(i\ne j)
\end{equation}
(Gil, eq.~(7)), a real antisymmetric matrix; when some populations vanish,
$M$ is defined on the support $\{i:a_i>0\}$ and extended by zeros to the
inactive axes (Gil, Section 2.3), so its entries on inactive axes are not
free variables. Then
\begin{equation}\label{eq:factor}
\rho=A^{1/2}(I+iM)A^{1/2}
\end{equation}
(Gil, eq.~(8)). Since $iM$ is Hermitian with eigenvalues $\pm s_k$ (and $0$
when $n$ is odd), where
$s_1\ge s_2\ge\dots\ge s_m\ge0$ ($m=\lfloor n/2\rfloor$) are the Youla values
of $M$, positivity of $\rho$ is equivalent to $s_1\le1$, that is, to
\begin{equation}\label{eq:contraction}
\|M\|_\op\le1 .
\end{equation}
Gil calls $\|N\|_\F$ the cohesion of the antisymmetric block; its square is
\begin{equation}\label{eq:cohesion}
\|N\|_\F^2=2\sum_{i<j}a_ia_jM_{ij}^2
\end{equation}
(Gil, eqs.~(20) and (33)), and it enters his complementarity identities for
purity, spectral sharpening and R\'enyi entropies. By the real normal form of
a real skew-symmetric matrix under orthogonal congruence (Youla~\cite{youla}
for the complex case under unitary congruence; \cite[Section~2.5]{hjma} for the
real case) there is $Q_M\in\Orth(n)$ with
$Q_M^TMQ_M=\bigoplus_k\left(\begin{smallmatrix}0&s_k\\-s_k&0\end{smallmatrix}\right)\oplus0$;
Gil calls $\rho$ aligned (his Definition 1) when, in some admissible IRB
representative, $Q_M$ can be taken to be a signed permutation of the IRB axes,
so that $M$ is supported on a matching of index pairs.

\begin{theorem}[Gil~\cite{gil}, Theorem 2]\label{thm:gil}
For fixed ordered populations, the maximum of $\|N\|_\F^2$ over the aligned
class is $2\sum_{k=1}^{m}a_{2k-1}a_{2k}$, attained by the adjacent pairing
$(1,2),(3,4),\dots$ with $s_k=1$ on every pair of positive population product.
\end{theorem}

Gil then writes, in the paragraph after his Theorem 2 (p.~10 of the publisher's PDF, where the section is numbered 3.5; the PubMed Central HTML rendering numbers it 3.4): ``Whether the same maximum is globally optimal
over arbitrary Youla orientations is a separate optimization problem and is
not assumed here,'' and lists in Section 8.2 three problems ``deliberately
left open because they require optimization over Grassmannian orbits rather
than the elementary matching argument used here''; they are unnumbered there,
and we number them (i) to (iii): (i) to determine the global maximum of
$\|N\|_\F^2$ for fixed populations and fixed Youla values over all
orientations $Q_M$; (ii) to characterize the image of the population-weighted
Gram map of Pl\"ucker coordinates; (iii) to decide whether the adjacent-pairing
maximum of Theorem~\ref{thm:gil} remains globally optimal beyond the aligned
class under suitable spectral constraints.

We determine the maximum asked for in (i) (Theorem~\ref{thm:prescribed}) and
answer (iii) affirmatively (Theorem~\ref{thm:free}); (ii) is not addressed. In
(iii) we read ``suitable spectral constraints'' as constraints on the Youla
values: if the spectrum of $\rho$ were fixed together with the populations,
$\tr\rho^2=\sum_ia_i^2+\|N\|_\F^2$ would already fix the objective, and the
remaining question, whether an aligned representative is compatible with such
a restriction on the spectrum of $\rho$, would be a feasibility question that
we do not address. For free Youla values the Grassmannian orbit never has to be parametrized:
the squared entries of every orientation lie in Edmonds' matching polytope,
so the maximization becomes linear optimization over a polytope whose
vertices are matchings. For prescribed values the product-weight maximum
follows instead from the singular-value inequality of
Theorem~\ref{thm:prescribed}.

\section{Squared entries of an antisymmetric contraction}

Let $\cM_n$ be the set of matchings of $\{1,\dots,n\}$ (sets of disjoint
pairs, the empty matching included), $\mathbf1_\mu\in\mathbb R^{\binom n2}$ the
incidence vector of a matching $\mu$ (the letter $M$ is reserved for the
metaspin tensor), $E(B)$ the pairs inside a vertex set $B$, and $\delta(i)$
the pairs containing $i$. Edmonds~\cite{edmonds} proved that the convex hull
of the incidence vectors is
\begin{equation}\label{eq:edmonds}
\cP_n=\Big\{x\ge0:\ \sum_{e\in\delta(i)}x_e\le1\ (1\le i\le n),\quad
\sum_{e\in E(B)}x_e\le\tfrac{|B|-1}2\ (|B|\ \text{odd})\Big\}.
\end{equation}

\begin{lemma}[squared-entry hull, from \cite{ross}]\label{lem:hull}
Let $K$ be an $n\times n$ real or complex skew-symmetric matrix with
$\|K\|_\op\le1$ and put $x_{\{i,j\}}=|K_{ij}|^2$ for $i<j$. Then $x\in\cP_n$.
Conversely every $\mathbf1_\mu$ is of this form, so the convex hull of the set of
such vectors is exactly $\cP_n$.
\end{lemma}

\begin{proof}
Nonnegativity is clear. The $i$-th row of a contraction has squared norm at
most $1$, which gives the degree inequalities. For $|B|$ odd, the principal
submatrix $K_B$ is skew-symmetric of odd order, so
$\det K_B=\det K_B^T=\det(-K_B)=-\det K_B$, hence $\det K_B=0$ and
$\rank K_B\le|B|-1$. $K_B$ is a contraction, so each of its singular values is
at most $1$ and $2\sum_{e\in E(B)}x_e=\|K_B\|_\F^2\le\rank K_B\le|B|-1$. Thus
$x$ satisfies \eqref{eq:edmonds}. Conversely, the matrix with block
$\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)$ on each pair of
$\mu$ and zeros elsewhere is a skew-symmetric partial isometry, hence a
contraction, with squared entries $\mathbf1_\mu$. Since $\cP_n$ is convex and
contains all the vectors, and its vertices are among them, the hull identity
follows.
\end{proof}

This lemma is taken from the companion manuscript~\cite{ross}, where it
drives an exact reduction of approximate antiunitary symmetry to weighted
matching; its application to density matrices was found independently by
both reviews of that manuscript (its Section~1 says so) and is developed
here. The set of squared-entry
vectors itself is not convex (a strict mixture of two matchings need not be
realized), and no point violating an odd-set inequality is realized by any
contraction; the hull statement is what matters below.

\section{Free Youla values}

\begin{theorem}\label{thm:free}
Let $a_1\ge\dots\ge a_n\ge0$ with $\sum_ia_i=1$ be fixed. Over all density matrices whose
intrinsic representation \eqref{eq:irb} has these populations, with every
orientation of the Youla frame and every admissible Youla spectrum,
\begin{equation}\label{eq:free}
\max\|N\|_\F^2=2\sum_{k=1}^{m}a_{2k-1}a_{2k},
\end{equation}
attained inside the aligned class by the adjacent pairing with $s_k=1$ on
every adjacent pair of positive population product. More generally, if all populations are positive, then for any
nonnegative weights $b_{ij}$,
\begin{equation}\label{eq:weights}
\max\sum_{i<j}b_{ij}M_{ij}^2=\max_{\mu\in\cM_n}\sum_{\{i,j\}\in\mu}b_{ij},
\end{equation}
the maximum weight of a matching, and the set of admissible vectors
$(M_{ij}^2)_{i<j}$ has convex hull $\cP_n$; if some populations vanish, both
statements hold with the matchings and the polytope of the support of $A$.
\end{theorem}

\begin{proof}
Suppose first that all populations are positive. By \eqref{eq:factor} and
\eqref{eq:contraction}, the admissible coherence blocks are exactly
$N=A^{1/2}MA^{1/2}$ with $M$ a real antisymmetric contraction, and by
\eqref{eq:cohesion} the squared cohesion is $2\sum_{i<j}a_ia_jx_{ij}$ with
$x=(M_{ij}^2)$. By Lemma~\ref{lem:hull}, $x\in\cP_n$. A linear functional with
nonnegative coefficients attains its maximum over the polytope at a vertex
$\mathbf1_\mu$, with value $2\sum_{\{i,j\}\in\mu}a_ia_j$. Among matchings the
adjacent pairing is optimal. First complete $\mu$ to a matching of maximal
size, which does not decrease the weight since the weights are nonnegative.
If $n$ is odd and the unpaired vertex is $v<n$, give $v$ the partner of $n$
and leave $n$ unpaired; the weight does not decrease since $a_v\ge a_n$. Now,
for $a_i\ge a_j\ge a_k\ge a_l$,
\begin{align*}
a_ia_j+a_ka_l-(a_ia_k+a_ja_l)&=(a_i-a_l)(a_j-a_k)\ge0,\\
a_ia_j+a_ka_l-(a_ia_l+a_ja_k)&=(a_i-a_k)(a_j-a_l)\ge0,
\end{align*}
so any crossing or nested pair of pairs can be uncrossed without decreasing
the weight; these are the exchange inequalities (26), (27) of~\cite{gil}. Each
uncrossing of pairs $\{i,k\},\{j,l\}$ or $\{i,l\},\{j,k\}$ with $i<j<k<l$ into
$\{i,j\},\{k,l\}$ lowers $\sum_{\{i,j\}\in\mu}|i-j|$ by $2(k-j)>0$, so the
process stops, and the only perfect matching of $\{1,\dots,2m\}$ without a
crossing or nested pair is the adjacent pairing. The adjacent signed swap with
$s_k=1$ is an antisymmetric contraction, so the value is attained, and the
corresponding state is aligned; explicitly, it is block diagonal with the
blocks
\[
\begin{pmatrix}a_{2k-1}&i\sqrt{a_{2k-1}a_{2k}}\\-i\sqrt{a_{2k-1}a_{2k}}&a_{2k}\end{pmatrix},
\]
of eigenvalues $a_{2k-1}+a_{2k}$ and $0$, followed by the unpaired population
when $n$ is odd: Hermitian, positive semidefinite, of trace one, with real
part $A$ and $\|N\|_\F^2=2\sum_ka_{2k-1}a_{2k}$. The same argument gives \eqref{eq:weights}
for arbitrary nonnegative weights, and the hull statement is the second half
of Lemma~\ref{lem:hull}. If some populations vanish, positivity forces the
corresponding rows and columns of $N$ to vanish (\cite{gil}, Section 2.3), so
$M$ is defined on the support of $A$ and the argument applies there; the zero
populations contribute nothing to \eqref{eq:free}, in agreement with the
formula, while \eqref{eq:weights} and the hull statement must be read on the
support: for $a=(\tfrac12,\tfrac12,0,0)$, $b_{12}=1$, $b_{34}=10$, the
maximum over states is $1$ and the unrestricted matching weight is $11$.
\end{proof}

Theorem~\ref{thm:free} answers question (iii): under the only constraint Gil
imposes, positivity, the adjacent-pairing maximum of Theorem~\ref{thm:gil} is
the global maximum, and the aligned class contains a maximizer of every
nonnegative pairwise coherence functional. The value \eqref{eq:free} also
follows from Mathias's inequality with $\|M\|_\op\le1$ (Remark~\ref{rem:lit});
what the matching polytope adds is the general-weight statement
\eqref{eq:weights} and the hull.

\section{Prescribed Youla values}

The inequality below follows from Horn's product inequality~\cite{horn50}
and is the Frobenius case of a theorem of Mathias~\cite{mathias}, known to
us through its zbMATH review; Remark~\ref{rem:lit} gives both deductions.

\begin{theorem}\label{thm:prescribed}
Let $a_1\ge\dots\ge a_n\ge0$, $m=\lfloor n/2\rfloor$, and $s_1\ge\dots\ge s_m\ge0$.
Let $M$ be any real or complex skew-symmetric $n\times n$ matrix whose singular
values are $s_1,s_1,\dots,s_m,s_m$ (and $0$ if $n$ is odd); equivalently
$M=Q\Sigma_sQ^T$ with $Q$ orthogonal or unitary and $\Sigma_s$ the Youla block
form. Then
\begin{equation}\label{eq:prescribed}
\sum_{i<j}a_ia_j|M_{ij}|^2\ \le\ \sum_{k=1}^{m}s_k^2\,a_{2k-1}a_{2k},
\end{equation}
with equality for $Q=I$. Consequently, for fixed populations and fixed Youla
values with $s_1\le1$, the global maximum of $\|N\|_\F^2$ over all
orientations $Q_M$ is $2\sum_ks_k^2a_{2k-1}a_{2k}$, attained in the aligned
class by placing the $k$-th largest Youla value on the $k$-th adjacent
population pair. (For positive populations every orientation with $s_1\le1$
is realized by a state. If $r<n$ populations are positive, a state exists only
when at most $\lfloor r/2\rfloor$ of the $s_k$ are positive, and the
statement is read on the support of $A$.)
\end{theorem}

\begin{proof}
Write $x_{ij}=|M_{ij}|^2$. For nonnegative reals,
$a_ia_j=\int_0^\infty\!\!\int_0^\infty\mathbf1[a_i>u]\,\mathbf1[a_j>v]\,du\,dv$,
and averaging this with the same identity in the opposite order,
\begin{equation}\label{eq:layer}
\sum_{i<j}a_ia_jx_{ij}=\iint\Psi(u,v)\,du\,dv,\qquad
\Psi(u,v)=\tfrac12\sum_{i<j}x_{ij}\Big(\mathbf1[a_i>u,\,a_j>v]+\mathbf1[a_i>v,\,a_j>u]\Big).
\end{equation}
Likewise the right side of \eqref{eq:prescribed} equals $\iint h(u,v)\,du\,dv$
with $h(u,v)=\frac12\sum_ks_k^2(\mathbf1[a_{2k-1}>u,a_{2k}>v]+\mathbf1[a_{2k-1}>v,a_{2k}>u])$.
Both integrands are symmetric in $(u,v)$, so it suffices to prove
$\Psi(u,v)\le h(u,v)$ when $u\ge v$. Fix such $(u,v)$ and let
$C=\{i:a_i>u\}$ and $B=\{i:a_i>v\}$. Because $a$ is sorted these are prefix
sets, $C=\{1,\dots,p\}\subseteq B=\{1,\dots,q\}$.

\emph{The integrand $\Psi$.} For $i<j$ we have $a_i\ge a_j$, hence
$\mathbf1[a_i>u,a_j>v]=\mathbf1[i\in C,\,j\in B]$ and
$\mathbf1[a_i>v,a_j>u]=\mathbf1[j\in C]$ (if $j\in C$ then $i\in C\subseteq B$).
The pairs of $B$ whose smaller index lies in $C$ are the pairs inside $C$
together with the pairs between $C$ and $B\setminus C$, so
\[
2\Psi(u,v)=2\,x(E(C))+x(C,B\setminus C),
\]
where $x(C,B\setminus C)$ denotes the mass on pairs with one end in $C$ and
the other in $B\setminus C$. In matrix terms $2x(E(C))=\|M_C\|_\F^2$ and
$x(C,B\setminus C)=\|M_{C,B\setminus C}\|_\F^2$, whence
\begin{equation}\label{eq:psi}
\Psi(u,v)=\tfrac12\|M_{C,B}\|_\F^2,
\end{equation}
one-half the squared Frobenius norm of the $p\times q$ block of $M$ with rows
$C$ and columns $B$.

\emph{The integrand $h$.} $\mathbf1[a_{2k-1}>u,a_{2k}>v]=\mathbf1[2k-1\le p,\,2k\le q]$
and $\mathbf1[a_{2k-1}>v,a_{2k}>u]=\mathbf1[2k\le p]$. Counting,
\begin{equation}\label{eq:h}
h(u,v)=\tfrac12R(p,q),\qquad
R(p,q)=\sum_{k\le\min(\lceil p/2\rceil,\lfloor q/2\rfloor)}s_k^2+\sum_{k\le\lfloor p/2\rfloor}s_k^2 .
\end{equation}

\emph{The inequality $\|M_{C,B}\|_\F^2\le R(p,q)$.} The block $M_{C,B}$
consists of $p$ rows of the principal submatrix $M_B$, which is
skew-symmetric of order $q$; its singular values come in pairs
$\tau_1\ge \tau_1\ge \tau_2\ge \tau_2\ge\dots\ge \tau_r\ge \tau_r$, $r=\lfloor q/2\rfloor$, with an
additional $0$ if $q$ is odd. Two classical facts apply. First, for any
matrix $X$ and the coordinate projection $P$ onto $p$ of its rows,
$\|PX\|_\F^2=\tr(PXX^*P)\le\sum_{k\le p}\sigma_k(X)^2$ (Ky Fan's maximum
principle~\cite{fan,bhatia}). Second, the singular values of a submatrix are
dominated termwise by those of the matrix, $\sigma_k(M_B)\le\sigma_k(M)$
(\cite[Thm.~3.1.2, Cor.~3.1.3]{hornjohnson}), which, both sequences being
sorted decreasingly, gives $\tau_k\le s_k$.
Hence $\|M_{C,B}\|_\F^2\le\sum_{k\le p}\sigma_k(M_B)^2$, and we compare this
sum with $R(p,q)$ in the three possible cases.
\begin{itemize}
\item $p$ even: $p/2\le\lfloor q/2\rfloor$, so
$\sum_{k\le p}\sigma_k(M_B)^2=2\sum_{k\le p/2}\tau_k^2\le2\sum_{k\le p/2}s_k^2=R(p,q)$.
\item $p$ odd and $p<q$: then $(p+1)/2\le\lfloor q/2\rfloor$, so
$\sum_{k\le p}\sigma_k(M_B)^2=2\sum_{k\le(p-1)/2}\tau_k^2+\tau_{(p+1)/2}^2
\le2\sum_{k\le(p-1)/2}s_k^2+s_{(p+1)/2}^2=R(p,q)$.
\item $p=q$ odd: $\sum_{k\le p}\sigma_k(M_B)^2=2\sum_{k\le r}\tau_k^2\le2\sum_{k\le(q-1)/2}s_k^2=R(p,q)$.
\end{itemize}
Thus $\Psi\le h$ pointwise, and integrating \eqref{eq:layer} proves
\eqref{eq:prescribed}. For $Q=I$ the only nonzero entries are
$M_{2k-1,2k}=-M_{2k,2k-1}=s_k$, and the left side equals the right side.
The consequence for the squared cohesion follows from \eqref{eq:cohesion} and
the fact that the aligned configuration is admissible; for the density-matrix
equality case replace the off-diagonal entries of the blocks in the proof of
Theorem~\ref{thm:free} by $\pm is_k\sqrt{a_{2k-1}a_{2k}}$, whose determinant
$a_{2k-1}a_{2k}(1-s_k^2)$ is nonnegative.
\end{proof}

\begin{proof}[Second proof, from a classical singular-value inequality]
Put $D=\diag(\sqrt{a_1},\dots,\sqrt{a_n})$ and $T=DMD$. Then $T$ is
skew-symmetric, so its singular values are $t_1,t_1,\dots,t_m,t_m$ (and $0$
when $n$ is odd), and
$\sum_{i<j}a_ia_j|M_{ij}|^2=\tfrac12\|T\|_\F^2=\sum_kt_k^2$. Horn's
inequality for the singular values of a product~\cite{horn50} states that
$\prod_{j\le r}\sigma_j(XY)\le\prod_{j\le r}\sigma_j(X)\sigma_j(Y)$ for every
$r$. Applied twice at the even index $r=2\ell$, with $\sigma_j(D)=\sqrt{a_j}$
in decreasing order,
\[
\prod_{k\le\ell}t_k^2=\prod_{j\le2\ell}\sigma_j(DMD)
\le\Big(\prod_{j\le2\ell}\sigma_j(D)^2\Big)\prod_{j\le2\ell}\sigma_j(M)
=\prod_{k\le\ell}s_k^2a_{2k-1}a_{2k}\qquad(1\le\ell\le m).
\]
Both $(t_k^2)_k$ and $(s_k^2a_{2k-1}a_{2k})_k$ are nonincreasing, so this says
that the first is weakly log-majorized by the second, which implies weak
majorization (apply the increasing convex function $\exp$ to the logarithms;
zeros are handled by continuity), and in particular
$\sum_kt_k^2\le\sum_ks_k^2a_{2k-1}a_{2k}$, which is \eqref{eq:prescribed}.
Equality for $Q=I$ is as before.
\end{proof}

Theorem~\ref{thm:prescribed} answers question (i), and the second proof shows
that, as a matrix inequality, it is a sharp consequence of classical
majorization, and indeed a special case of Mathias's 1992 inequality
(Remark~\ref{rem:lit}); what the layer-cake proof adds is the family of
Proposition~\ref{prop:family}. The theorem recovers \eqref{eq:free}: apply it
to each contraction with its own Youla values and use that the right side of
\eqref{eq:prescribed} is nondecreasing in every $s_k$. It does not recover
\eqref{eq:weights} or the hull statement, which concern non-product weights;
Remark~\ref{rem:nonproduct} shows that the prescribed-value analog of
\eqref{eq:weights} is false, so they cannot be obtained one Youla spectrum at
a time.

\begin{proposition}[valid inequalities for prescribed spectrum]\label{prop:family}
Let $M$ be skew-symmetric with singular values $s_1,s_1,\dots,s_m,s_m$ (and
$0$ if $n$ is odd), and $x_{ij}=|M_{ij}|^2$. For all vertex sets $C\subseteq B$,
\begin{equation}\label{eq:family}
2\sum_{e\in E(C)}x_e+\sum_{e\in E(B)\setminus E(C)\setminus E(B\setminus C)}x_e\ \le\ R(|C|,|B|),
\end{equation}
with $R$ as in \eqref{eq:h}. For $|C|=1$ and $|B|\ge2$ this is the degree constraint
$\sum_{e\in\delta(i)\cap E(B)}x_e\le s_1^2$ (the singleton $C=B$ reads $0\le0$); for $C=B$ it is the refined
odd/even-set constraint $\sum_{e\in E(B)}x_e\le\sum_{k\le\lfloor|B|/2\rfloor}s_k^2$,
which for $s\equiv1$ is Edmonds' odd-set inequality.
\end{proposition}

\begin{proof}
The proof of Theorem~\ref{thm:prescribed} never used that $C$ and $B$ are
prefix sets: the left side of \eqref{eq:family} is $\|M_{C,B}\|_\F^2$ and the
bound is the one just established.
\end{proof}

\begin{remark}\label{rem:lp}
The subfamily $C=B$ of \eqref{eq:family} is not sufficient for
Theorem~\ref{thm:prescribed}: the linear program that maximizes
$\sum_{i<j}a_ia_jx_{ij}$ subject only to those constraints exceeds the true
value of that sum in random instances (by up to $0.0097$ in the checks of
Section~\ref{sec:checks}, that is, $0.019$ in units of $\|N\|_\F^2$), and
adding the degree constraints to $C=B$ does not close the gap (in $4$ of the
$30$ instances the value is still exceeded, by up to $2.2\cdot10^{-4}$, or
$4.4\cdot10^{-4}$ in units of $\|N\|_\F^2$). Conversely, the proof uses only the
$n(n+1)/2$ prefix inequalities $C=\{1,\dots,p\}\subseteq B=\{1,\dots,q\}$, and
the linear program over those alone reproduces the value of
Theorem~\ref{thm:prescribed} to machine precision in every instance, as does
the linear program over the full family \eqref{eq:family}. Some mixed
constraints with $C\subsetneq B$ are therefore essential, and the prefix ones
suffice.
\end{remark}

\begin{remark}\label{rem:lit}
The zero diagonal of a skew matrix is what allows the product form. The
generic bound for a Hermitian matrix $X$ with eigenvalues $\mu_i$,
$\|A^{1/2}XA^{1/2}\|_\F^2=\tr(AXAX)\le\tr(A^2X^2)\le\sum_ia_i^2\mu_i^{2}$
(Cauchy--Schwarz, then von Neumann's trace inequality with both sequences
sorted), applied to $X=iM$ gives only $\sum_ks_k^2(a_{2k-1}^2+a_{2k}^2)$,
which is weaker than \eqref{eq:prescribed} by the arithmetic--geometric mean
inequality and is attained by diagonal $X$, a form unavailable to $iM$.
As a matrix inequality, \eqref{eq:prescribed} is not new in substance: the
second proof of Theorem~\ref{thm:prescribed} derives it in a few lines from
Horn's 1950 product inequality~\cite{horn50}, and it is a special case of a
theorem of Mathias found after this note was written and known to us through
its zbMATH review (Zbl 0759.15011; the paper itself was not obtained), as
follows. The skew-symmetric Schur--Horn
theorem (Leite, Richa and Tomei~\cite{lrt}; \cite[Prop.~3.10]{sss}, which
concerns the linear skew-diagonal projection of an orbitope) and the main
theorem of Tam~\cite{tam} describe the entries on one fixed matching under
congruence, the moduli of the skew-diagonal entries being weakly majorized by
the Youla values (with one further inequality, of Thompson--Sing type, in
Tam's even-dimensional full-matching case); they do not
directly state the weighted squared-entry inequality. Mathias~\cite{mathias}
proved, as the review states it, that for a real positive semidefinite $A$
with ordered diagonal entries $\alpha_1\ge\dots\ge\alpha_n$ and a complex
skew-symmetric $B$,
$\sum_{i\le k}\sigma_i(A\circ B)\le\sum_{i\le k}\tilde\alpha_i\,\sigma_i(B)$
for every $k$, where $\tilde\alpha_{2i-1}=\tilde\alpha_{2i}=\sqrt{\alpha_{2i-1}\alpha_{2i}}$
(and $\tilde\alpha_n=0$ for odd $n$), together with the corresponding bound in
every unitarily invariant norm. With $A=dd^T$, $d_i=\sqrt{a_i}$, the Hadamard
product is exactly the matrix $DMD=(dd^T)\circ M$ of the second proof, whose
singular values are $t_1,t_1,t_2,t_2,\dots$, so his inequality says that
$(t_k)_k$ is weakly majorized by $(\sqrt{a_{2k-1}a_{2k}}\,s_k)_k$, and the
convexity of the square gives $\sum_kt_k^2\le\sum_ks_k^2a_{2k-1}a_{2k}$,
which is \eqref{eq:prescribed}; with $\|M\|_\op\le1$ it also gives the value
\eqref{eq:free}. We claim no priority for \eqref{eq:prescribed}: it is the
Frobenius case of Mathias's theorem. Tam and Leite--Richa--Tomei were
available to us only through their abstracts (Sanyal--Sottile--Sturmfels in
full), and we do not claim that their papers contain no equivalent
formulation. This note's contribution is the application to Gil's questions, through
the hull lemma of the companion manuscript, and the nested-set family of
Proposition~\ref{prop:family}; the searches described here do not establish
priority for these formulations either.
\end{remark}

\begin{remark}[product weights are essential]\label{rem:nonproduct}
With prescribed Youla values and general nonnegative weights $b_{ij}$ the
maximum of $\sum_{i<j}b_{ij}M_{ij}^2$ need not be attained in the aligned
class, and the linear program over \eqref{eq:family} need not be tight. Take
$n=4$ with distinct positive populations, say $a=(0.4,0.3,0.2,0.1)$, so that
alignment means matching support (distinct populations leave no residual
rotation in Gil's Definition 1), $s=(1,0.9)$, $b_{12}=2$, $b_{13}=b_{24}=1.2$,
all other $b_{ij}=0$, and $c=\sqrt{0.84}$. The real skew matrix with
$M_{12}=0.3$, $M_{13}=c$, $M_{24}=-c$, $M_{34}=0.2$ (and zeros elsewhere above
the diagonal) has $s_1^2+s_2^2=\tfrac12\|M\|_\F^2=1.81$ and
$s_1s_2=|\operatorname{Pf}M|=0.9$, so its singular values are $1,1,0.9,0.9$;
its value is $2(0.09)+1.2(0.84)+1.2(0.84)=2.196$, while the best aligned value
is $2.172$ (the matching $\{13,24\}$ with $s_1^2,s_2^2$ in either order;
$\{12,34\}$ gives $2.0$ and $\{14,23\}$ gives $0$). The linear program over
\eqref{eq:family} gives $2.324$: the point $x_{12}=0.19$, $x_{13}=x_{24}=0.81$,
all other $x_e=0$, is feasible with objective $2.324$, and the combination of
$0.6$ times the inequality for $C=\{1,2\}$, $B=[4]$ with $0.2$ times each of
the inequalities for $C=\{1,2,3\}$ and $C=\{1,2,4\}$, $B=[4]$, bounds the
objective by $0.6\cdot2+0.2\cdot2.81+0.2\cdot2.81=2.324$. The value $2.196$
is the exact maximum over $\Orth(4)$: with
$X=\tfrac12(M_{12}+M_{34},\,M_{13}-M_{24},\,M_{14}+M_{23})$ and
$Y=\tfrac12(M_{12}-M_{34},\,M_{13}+M_{24},\,M_{14}-M_{23})$ (in the notation
of Gil's Appendix~A, $X=\tfrac12A_+\hat v_+$ and $Y=\tfrac12A_-\hat v_-$ with
$A_\pm=s_1\pm s_2$ on one orientation class \cite[eqs.~(A3)--(A6)]{gil}) the
constraints
$\tfrac14\|M\|_\F^2=\|X\|^2+\|Y\|^2=0.905$ and
$|\operatorname{Pf}M|=\bigl|\|X\|^2-\|Y\|^2\bigr|=0.9$ (an orientation-reversing
orthogonal congruence changes the sign of the Pfaffian and exchanges the
roles of $X$ and $Y$, so only the absolute value is invariant on the orbit)
give $\{\|X\|^2,\|Y\|^2\}=\{0.9025,0.0025\}$, the objective is
$2(X_1+Y_1)^2+2.4(X_2^2+Y_2^2)$, symmetric in $X$ and $Y$, and when
$\|Y\|^2=0.0025$ it is at most
$2.172-0.4(X_1-5Y_1)^2+9.6Y_1^2\le2.172+9.6\cdot0.0025=2.196$, with equality
at the displayed matrix. This note does not determine the maximum for general weights.
\end{remark}

\section{Consequences for the intrinsic description}

\begin{corollary}\label{cor:consequences}
Let the populations be fixed.
\begin{enumerate}
\item With free Youla values, every functional of the form
$\sum_{i<j}b_{ij}M_{ij}^2$ with $b_{ij}\ge0$ is maximized by an aligned state
whose support is a maximum-weight matching for the weights $b$ on the support
$\{i:a_i>0\}$ (every entry on an inactive axis is zero, so for positive
populations the matching is on all of $[n]$); in particular Gil's
hierarchical descriptors that are linear in the squared metaspin entries are
extremized in the aligned class.
\item For positive populations, the set of achievable coherence weight vectors
$(N_{ij}^2/(a_ia_j))_{i<j}$ over all states with these populations has convex
hull equal to Edmonds' matching polytope $\cP_n$; in particular no state can
put total squared metaspin weight more than $(|B|-1)/2$ on the pairs inside an
odd set $B$, whatever its orientation.
\item With Youla values also fixed ($s_1\le1$, read on the support of the
populations), the squared cohesion is at most
$2\sum_ks_k^2a_{2k-1}a_{2k}$, and the bound is attained only if
$\|M_{[p],[q]}\|_\F^2=R(p,q)$ for every $p\le q$ with $a_p>a_{p+1}$ and
$a_q>a_{q+1}$ (with $a_{n+1}=0$), that is, only if every prefix inequality
used in the proof is tight on the level sets of the populations. Whether this
forces alignment is not examined here.
\end{enumerate}
\end{corollary}

\begin{proof}
(1) and (2) are Theorem~\ref{thm:free} and Lemma~\ref{lem:hull}; (3) is
Theorem~\ref{thm:prescribed} together with the observation that equality in
the integrated inequality forces equality in $\Psi\le h$ for almost every
$(u,v)$, and the prefix pairs $(C,B)$ that occur for a set of $(u,v)$ of
positive measure are exactly those named.
\end{proof}

The results here do not determine the image of Gil's Gram map $G(a,Q_M)$
(his question (ii)), or the maximum of a general nonnegative-weight
functional $\sum_{i<j}b_{ij}M_{ij}^2$ with prescribed Youla values;
Remark~\ref{rem:nonproduct} shows that alignment and the linear program over
\eqref{eq:family} do not solve the latter in general.

\section{Numerical checks}\label{sec:checks}

The program \texttt{verify\_gil.py} (seed $20260929$; results in
\texttt{verification-gil.json}; both published with this note at
\url{https://hypnosmath.org/research/global-maximal-coherence-two-questions-of-gil}) performs
the following finite checks. They
are implementation controls, not part of the proofs; the searches are local,
and the program records the worst shortfall of each search as well as its
largest excess, so that a search that fails to reach a bound is visible.
\begin{itemize}
\item Free Youla values, $2\le n\le7$, five population vectors per $n$ ($30$
instances): $300$ random antisymmetric contractions each, and projected
gradient ascent over the contraction ball ($20$ random starts, $4000$ steps,
the singular values clipped at $1$ after each step, a final snap to a partial
isometry); the squared cohesion never exceeded \eqref{eq:free} beyond roundoff
(largest excess $5.6\cdot10^{-16}$), the ascent came within $10^{-6}$ of it in
$19$ instances and within $10^{-4}$ in all $30$, and its worst shortfall was
$7.2\cdot10^{-5}$ (at $n=7$).
\item Prescribed Youla values, same dimensions, five spectra per $n$ including
$s\equiv1$: $300$ random orientations and $20$ Nelder--Mead maximizations,
both over $\mathrm{SO}(n)$ through the exponential parametrization (a
coordinate sign flip preserves squared entries, so this covers $\Orth(n)$);
the value $2\sum_ks_k^2a_{2k-1}a_{2k}$ was not exceeded beyond numerical error
(largest excess $1.5\cdot10^{-13}$) and was reached within $10^{-6}$ in all
$30$ instances.
\item Linear programs over the full family \eqref{eq:family}, over the prefix
subfamily alone, over $C=B$ alone, and over $C=B$ with the degree constraints,
compared with Theorem~\ref{thm:prescribed}: the first two agree with the
theorem to $5.6\cdot10^{-17}$ in every instance; the last two exceed it as
reported in Remark~\ref{rem:lp}.
\item Floating-point checks of the odd-set inequalities for random
antisymmetric contractions on every odd subset, $3\le n\le8$ ($8760$
checks, largest excess $-0.032$).
\end{itemize}

\section{Provenance}\label{sec:provenance}

The two questions were found while reviewing the companion manuscript~\cite{ross}, whose hull
lemma this note reuses, and the theorems were proved in that session; no step of the mathematics
came from the harness. The companion manuscript arose from Hypnos~\cite{HypnosMethods}, a research harness built at the direction of David Ross. Its proposal models are Gemma~4 31B, a dense model in 4-bit activation-aware weight quantization (AWQ), \path{QuantTrio/gemma-4-31B-it-AWQ}; Gemma~4 26B, a mixture-of-experts model with 4 billion active parameters, served in AWQ from a quantization of \path{google/gemma-4-26B-A4B-it}; and Qwen3 32B, a dense model in 4-bit AWQ, \path{Qwen/Qwen3-32B-AWQ}. They run through vLLM on hardware operated by Ross and generate claims from a notebook of structured records. Claude Opus (Anthropic), accessed through a subscription, reviews samples under pre-registered rules; claims selected for execution include a computation that the harness runs. Proposals, verdicts and execution records are retained in the private repository and database. The site \url{https://hypnosmath.org} publishes reviewed explanations and selected computational records. The program of Section~\ref{sec:checks} and its
recorded outputs are published with this note at
\url{https://hypnosmath.org/research/global-maximal-coherence-two-questions-of-gil}, with the
note's source and its reviews.

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\end{document}