doc: include review comments

paper/shadow_maps_symmetric_states.tex

@@ -50,13 +50,20 @@ much finer block structure than a single collapsed correlation matrix.
 Several features of that structure --- the triple degeneracy of
 $\mathcal M_a(\GHZ_3)$'s singular values, the analogous triple
 degeneracy of the Smolin state under both the $1\mid3$ and $2\mid2$ cut,
-and the common saturating value $6/\sqrt7$ shared by all $38$ connected
+and the common saturating value $6/\sqrt7$ of the single-party ($1\mid3$)
+cut functionals $\Phi_{\mathrm{sym}}=\Phi_{\max}$ shared by all $38$ connected
 four-qubit graph states --- were reported there as separate numerical
 facts. We show these degeneracies are forced \emph{exactly}, by two
 structurally independent mechanisms. The first is representation-theoretic:
 whenever $\rho$ is invariant under a compact group acting by unitaries
 compatible with the cut, Schur's lemma forces $\mathcal M_S(\rho)$ to be
-block diagonal over isotypic components; we give the precise (weak,
+block diagonal over isotypic components -- with each diagonal block itself
+a scalar multiple of the identity on the real-type irreducible
+representations that occur throughout the $SO(3)$ and permutation
+mechanisms of this note (automatic there; the abelian $U(1)$ mechanism
+below is of complex, not real, type and is handled instead by
+complexifying and working with $1$-dimensional complex weight spaces
+directly, sidestepping rather than using this reduction); we give the precise (weak,
 cut-factorizing versus strong, party-local) hypotheses under which this
 is compatible with the existing sector grading, and exhibit an axial-$U(1)$
 four-qubit example, not itself a stabilizer state, where this mechanism
@@ -79,7 +86,22 @@ permutation group of its constituent parties: the corresponding
 shadow-map block reduces, without approximation, from a $3^m\times3^l$
 matrix to one of polynomial size $\binom{m+2}2\times\binom{l+2}2$, which
 we demonstrate for Dicke states with cluster sizes up to $m=40$, $l=35$
-within an $n=200$-qubit state.
+within an $n=200$-qubit state. We then work out the representation
+theory of the full, non-abelian collective hypothesis in detail: an
+exact branching formula for the $SU(2)$ multiplicities, exact isotypic
+projectors via the Casimir operator (no group averaging), an exact
+recoupling formula relating the reduced blocks of two different
+cuts of the same invariant tensor to one another by an explicit
+dimension-ratio factor -- proved in general except for one finite
+identity verified exhaustively and exactly for the six-qubit case
+treated here rather than in closed form for arbitrary cluster size --,
+a cut-independent template decomposition that
+lets every cut and every state in a fixed invariant subspace be
+recovered from $r^2$ fixed tensors without revisiting the full Hilbert
+space, and, combining this with the permutation collapse, a further
+multiplicity-free reduction to genuinely scalar reduced matrix elements
+on $O(m)$ channels -- all illustrated on explicit six-qubit examples and
+backed by exact symbolic and numerical verification throughout.
 \end{abstract}
 
 
@@ -194,6 +216,28 @@ clusters of up to $m=40$, $l=35$ qubits within an $n=200$-qubit state
 (Table~\ref{tab:dicke-scaling}), far beyond what the ambient matrix could
 represent explicitly.
 
+\emph{What if the collective symmetry is the full, non-abelian $SU(2)$,
+not just its abelian $U(1)$ subgroup?} Section~\ref{sec:full-su2-symmetry}
+develops this case in detail, going beyond the closed-form worked example
+of Section~\ref{sec:u1-example}: an exact branching formula for the
+resulting multiplicities (Proposition~\ref{prop:branching-su2}), exact
+isotypic projectors via the $J^2$ Casimir rather than group averaging
+(Corollary~\ref{cor:projector-norm-formula}), an exact recoupling
+formula relating the reduced blocks of two different cuts of the same
+invariant tensor by an explicit, closed-form factor -- proved in general
+except for one finite identity that is verified exhaustively and exactly
+for the six-qubit case treated here, rather than derived in closed form
+for arbitrary cluster size
+(Theorem~\ref{thm:six-j-recoupling}, Remark~\ref{rem:six-j-scope}), a cut-independent template
+decomposition letting every cut and every state in a fixed invariant
+subspace be recovered from finitely many fixed tensors without
+revisiting the full Hilbert space (Proposition~\ref{prop:coherence-templates}),
+and, combining this with the permutation collapse of
+Section~\ref{sec:polynomial-witness}, a further reduction to genuinely
+multiplicity-free, scalar reduced matrix elements
+(Section~\ref{sec:sm-so3-combination}) -- all illustrated on explicit
+six-qubit examples.
+
 Section~\ref{sec:setup} fixes notation and recalls, without reproof, the
 minimal apparatus from~\cite{aschauer2026a} needed below: the sector
 decomposition of $\V_0^{(S)}$, $\V_0^{(S^c)}$, the definition of
@@ -298,7 +342,7 @@ With $d_S:=\prod_{a\in S}d_a$ and $d_{S^c}:=\prod_{a\in S^c}d_a$, the
 \label{eq:bigraduated-map}
 \end{equation}
 i.e.\ the matrix of $\widetilde{\mathcal M}_S(\rho)$ in sector-adapted
-orthonormal coordinates on both sides. For a singleton source $S=\{a\}$,
+orthonormal coordinates on both sides. (Since the generators satisfy $\tr(\sigma_i\sigma_j)=d\,\delta_{ij}$, not $\delta_{ij}$, the raw $\sigma_i$ are Hilbert--Schmidt \emph{orthogonal} but not unit-normalized, so "orthonormal coordinates" means the basis $\{\sigma_i/\sqrt d\}$; the prefactor in Eq.~\eqref{eq:bigraduated-map} is exactly what converts matrix entries computed in the raw, unnormalized Pauli-string basis -- the basis in which every explicit computation and every worked example in this note, including the stabilizer mechanism of Section~\ref{sec:mechanisms} and all numerical scripts, is actually carried out -- into this orthonormal normalization.) For a singleton source $S=\{a\}$,
 the source-side decomposition~\eqref{eq:source-bloch-decomposition} has
 only the summand $V=S$, and \eqref{eq:bigraduated-map} reduces to the
 ordinary, target-only combined shadow map $\mathcal M_a(\rho)$ used
@@ -335,7 +379,9 @@ projections. If $\rho$ is separable across $S\mid S^c$, then
 \norm{P_{\mathcal T}\,\mathcal M_S(\rho)\,P_{\mathcal V}}_*\ \le\ \norm{\mathcal M_S(\rho)}_*\ \le\ 1,
 \end{equation*}
 with no separate proof required: orthogonal projections are contractive
-for the operator norm.
+for the operator norm ($\norm{P}_{\mathrm{op}}\le1$), and the nuclear norm
+is submultiplicative with respect to such contractions on each side,
+$\norm{AXB}_*\le\norm{A}_{\mathrm{op}}\norm{X}_*\norm{B}_{\mathrm{op}}$, giving the stated bound with $A=P_{\mathcal T}$, $X=\mathcal M_S(\rho)$, $B=P_{\mathcal V}$.
 \end{corollary}
 
 Everything in Sections~\ref{sec:mechanisms}--\ref{sec:polynomial-witness}
@@ -374,7 +420,7 @@ Under the weak, cut-factorizing hypothesis above, $\widetilde{\mathcal M}_S(\rho
 \Pi_\lambda^{(S^c)}\,\widetilde{\mathcal M}_S(\rho)\,\Pi_\mu^{(S)}=0
 \qquad\text{whenever }\lambda\neq\mu,
 \end{equation*}
-exactly, not merely approximately or in a bound. Moreover, on the surviving diagonal blocks,
+exactly, not merely approximately or in a bound. Moreover, whenever the real irreducible $V_\lambda$ is of \emph{real type} (Remark~\ref{rem:real-schur} below) -- the case for every representation actually used in this note -- the surviving diagonal blocks take the Wigner--Eckart form
 \begin{equation}
 \widetilde{\mathcal M}_S(\rho)\big|_\lambda = A_\lambda\otimes\mathrm{id}_{V_\lambda}
 \label{eq:wigner-eckart}
@@ -383,9 +429,14 @@ for a unique linear map $A_\lambda:M_\lambda^{(S)}\to M_\lambda^{(S^c)}$, the re
 \end{proposition}
 
 \begin{proof}
-$G$-invariance of $\rho$ gives $\mathrm{Ad}^{(S^c)}_g\,\widetilde{\mathcal M}_S(\rho)=\widetilde{\mathcal M}_S(\rho)\,\mathrm{Ad}^{(S)}_g$ for all $g$, i.e.\ $\widetilde{\mathcal M}_S(\rho)$ is a $G$-equivariant map between the two representations~\eqref{eq:isotypic-decomp}. Both claims are then Schur's lemma applied to the isotypic decomposition: equivariant maps vanish between inequivalent irreducible summands, and act as a fixed scalar multiple of the identity on the irreducible factor $V_\lambda$ within a matching pair, leaving exactly the freedom recorded in $A_\lambda$ on the multiplicity spaces.
+$G$-invariance of $\rho$ gives $\mathrm{Ad}^{(S^c)}_g\,\widetilde{\mathcal M}_S(\rho)=\widetilde{\mathcal M}_S(\rho)\,\mathrm{Ad}^{(S)}_g$ for all $g$, i.e.\ $\widetilde{\mathcal M}_S(\rho)$ is a $G$-equivariant map between the two representations~\eqref{eq:isotypic-decomp}. The block-vanishing claim is Schur's lemma applied to the isotypic decomposition: an equivariant map between real representations vanishes on isotypic components indexed by inequivalent irreducibles, with no further hypothesis needed. On a matching pair of real-type irreducible factors $V_\lambda$, the same equivariant map, restricted to $V_\lambda\to V_\lambda$, lies in $\mathrm{End}_G(V_\lambda)$, which by hypothesis is $\mathbb R\cdot\id$ (Remark~\ref{rem:real-schur}), giving Eq.~\eqref{eq:wigner-eckart} with $A_\lambda$ recording exactly the remaining freedom on the multiplicity spaces.
 \end{proof}
 
+\begin{remark}[Real, complex, and quaternionic type: where the real-type restriction is used, and where it is automatic]
+\label{rem:real-schur}
+Real Schur's lemma states only that $\mathrm{End}_G(V_\lambda)$ is a real division algebra for an irreducible real representation $V_\lambda$; by the Frobenius classification this is $\mathbb R$, $\mathbb C$, or $\mathbb H$, according to whether $V_\lambda$ is of real, complex, or quaternionic type (equivalently, Frobenius--Schur indicator $+1$, $0$, or $-1$). Only in the real-type case does an equivariant self-map of $V_\lambda$ reduce to a \emph{scalar} multiple of the identity as Eq.~\eqref{eq:wigner-eckart} asserts; in the complex or quaternionic case $A_\lambda$ would need to be valued in $\mathrm{Hom}_{\mathbb C}$ or $\mathrm{Hom}_{\mathbb H}$ of the respective multiplicity spaces, and Eq.~\eqref{eq:wigner-eckart} in its literal stated form does not hold. This restriction is genuinely inactive for two of the three symmetry mechanisms used in this note, but not the third, and it is worth being precise about which is which rather than claiming it away uniformly. Every representation appearing in Section~\ref{sec:full-su2-symmetry} is built entirely from $\mathcal V_0^{(a)}\cong\mathbb R^3$, the real (adjoint, spin-$1$) representation of $SO(3)$, and every irreducible integer-spin representation of $SO(3)$ is of real type (realizable over $\mathbb R$, e.g.\ via real spherical harmonics); half-integer spins, where the quaternionic case would arise, never occur in tensor powers of the spin-$1$ generator space, so Eq.~\eqref{eq:wigner-eckart} applies there exactly as stated. The permutation representations of Section~\ref{sec:mechanisms} used for $W_n$ and the ring/graph states are likewise real type (permutation matrices are real orthogonal, and the relevant irreducibles are the standard real irreducibles of $S_n$). The $U(1)$ mechanism of Section~\ref{sec:u1-example}, by contrast, is \emph{not} an instance of the real-type case: the real $2$-dimensional representation of $U(1)$ on the $(x,y)$-plane orthogonal to the rotation axis has $\mathrm{End}_{U(1)}(\mathbb R^2)\cong\mathbb C$ (generated by $\id$ and the rotation-by-$90^\circ$ matrix $J$ with $J^2=-\id$), i.e.\ it is of \emph{complex} type, and Eq.~\eqref{eq:wigner-eckart} does not apply to it in the literal real-scalar form stated above. Section~\ref{sec:u1-example} does not run into this obstruction because it does not use Eq.~\eqref{eq:wigner-eckart} at all for the $U(1)$ case: it complexifies first and works directly with the resulting pair of $1$-dimensional \emph{complex} weight spaces $E_+,E_-$ (mutually conjugate, of opposite charge), which is the standard and correct route for a complex-type real irreducible, and reduces the block structure to the elementary charge-conservation selection rule of Remark~\ref{rem:abelian-selection-rule} instead of a real Wigner-Eckart reduction. So the real-type restriction is inactive throughout Section~\ref{sec:full-su2-symmetry} and the permutation mechanism of Section~\ref{sec:mechanisms}, and is sidestepped rather than violated in Section~\ref{sec:u1-example}.
+\end{remark}
+
 Proposition~\ref{prop:block-diagonal} organizes $\widetilde{\mathcal M}_S(\rho)$ by symmetry channel alone, treating $\V_0^{(S)}$ and $\V_0^{(S^c)}$ as undifferentiated representations of $G$. But these spaces already carry the combinatorial sector grading of \cite{aschauer2026a}, $\V_0^{(S)}=\bigoplus_{\emptyset\neq V\subseteq S}\V_V^{(S)}$. The next result shows that this grading and the isotypic one are never in tension: for any symmetry acting locally by conjugation, one refines the other, and both can be imposed simultaneously without contradiction.
 
 \begin{proposition}[Compatibility of sector and symmetry decompositions]
@@ -483,7 +534,7 @@ By hypothesis $U_\pi\rho U_\pi^\dagger=\rho$, giving Eq.~\eqref{eq:transitive-eq
 
 \begin{proposition}[Reduced matrix element formula]
 \label{prop:reduced-formula}
-Let $\rho=T_G(\rho_S\otimes\sigma_{S^c})$ be the $G$-twirl of a product state across $S\mid S^c$, so $\rho$ is separable (a mixture, over $g\in G$, of product states) and $G$-invariant. Let $r^{(S)}\in\V_0^{(S)}$, $v_{S^c}\in\V_0^{(S^c)}$ be the traceless Bloch vectors of $\rho_S$, $\sigma_{S^c}$, with isotypic components $r_\lambda$, $v_\lambda$. Then
+Assume $V_\lambda$ is of real type (Remark~\ref{rem:real-schur}; automatic throughout this note). Let $\rho=T_G(\rho_S\otimes\sigma_{S^c})$ be the $G$-twirl of a product state across $S\mid S^c$, so $\rho$ is separable (a mixture, over $g\in G$, of product states) and $G$-invariant. Let $r^{(S)}\in\V_0^{(S)}$, $v_{S^c}\in\V_0^{(S^c)}$ be the traceless Bloch vectors of $\rho_S$, $\sigma_{S^c}$, with isotypic components $r_\lambda$, $v_\lambda$. Then
 \begin{equation}
 A_\lambda=\frac{1}{d_\lambda}\,\tilde v_\lambda\,\tilde r_\lambda^{\,T},
 \label{eq:reduced-element}
@@ -567,20 +618,20 @@ For each two-qubit cluster $S$ and $S^c$, the local weight decomposition $\{-1,0
 \toprule
 $q_S\to q_T$ & block size & singular value(s) & origin \\
 \midrule
-$+2\to-2$ & $1\times1$ & $\tfrac23\sin(2\varphi)$ & $E_+E_+\to E_-E_-$ \\
+$+2\to-2$ & $1\times1$ & $\tfrac23\lvert\sin(2\varphi)\rvert$ & $E_+E_+\to E_-E_-$ \\
 $+1\to-1$ & $2\times2$ & $0,0$ & identically zero for every $\varphi$ \\
 $0\to0$ & $3\times3$, rank $1$ & $\tfrac13$ & entirely from $E_0E_0\to E_0E_0$, i.e.\ $ZZ\to ZZ$ \\
 $-1\to+1$ & $2\times2$ & $0,0$ & identically zero for every $\varphi$ \\
-$-2\to+2$ & $1\times1$ & $\tfrac23\sin(2\varphi)$ & $E_-E_-\to E_+E_+$ \\
+$-2\to+2$ & $1\times1$ & $\tfrac23\lvert\sin(2\varphi)\rvert$ & $E_-E_-\to E_+E_+$ \\
 \bottomrule
 \end{tabular}
 \end{center}
-so that, exactly,
+so that, exactly and for every $\varphi\in\mathbb R$ (not merely $\varphi\in[0,\pi/2]$, where the entries $\tfrac23\sin(2\varphi)$ computed directly from the raw block happen already to be nonnegative),
 \begin{equation}
-\norm{\mathcal M_{AB}(\rho_\varphi)}_* = \frac13+\frac43\sin(2\varphi).
+\norm{\mathcal M_{AB}(\rho_\varphi)}_* = \frac13+\frac43\lvert\sin(2\varphi)\rvert.
 \label{eq:u1-example-formula}
 \end{equation}
-The $q=0$ value $\tfrac13$ is $\varphi$-independent, because $ZZZZ$ is diagonal in the computational basis and both terms of $\lvert\psi_\varphi\rangle$ give it the same eigenvalue $+1$; the entire $\varphi$-dependence, and with it the entanglement signal, sits in the $q=\pm2$ sectors. Equation~\eqref{eq:u1-example-formula} exceeds the cut-separable bound of Theorem~\ref{thm:cluster-cut} exactly when $\sin(2\varphi)>1/2$, i.e.\ for $\varphi\in(\pi/12,\,5\pi/12)$; it vanishes at $\varphi=0$, where $\rho_\varphi$ is itself a product state across $S\mid S^c$, and reaches $\norm{\mathcal M_{AB}(\rho_\varphi)}_*=5/3$ at the maximally superposed point $\varphi=\pi/4$.
+The two $q=\pm2$ singular values are genuinely $\lvert\tfrac23\sin(2\varphi)\rvert$, not $\tfrac23\sin(2\varphi)$ itself -- the raw $1\times1$ block entries computed from $\mathcal C(\rho_\varphi)$ can be negative, but a singular value cannot, and the nuclear norm sums the absolute values (verified directly against the true nuclear norm of the full raw $9\times9$ block for $\varphi$ outside $[0,\pi/2]$ in \texttt{scripts/check\_u1.py}). The $q=0$ value $\tfrac13$ is $\varphi$-independent, because $ZZZZ$ is diagonal in the computational basis and both terms of $\lvert\psi_\varphi\rangle$ give it the same eigenvalue $+1$; the entire $\varphi$-dependence, and with it the entanglement signal, sits in the $q=\pm2$ sectors. Equation~\eqref{eq:u1-example-formula} exceeds the cut-separable bound of Theorem~\ref{thm:cluster-cut} exactly when $\lvert\sin(2\varphi)\rvert>1/2$; restricting to $\varphi\in[0,\pi/2]$ for concreteness, this is $\varphi\in(\pi/12,\,5\pi/12)$, it vanishes at $\varphi=0$, where $\rho_\varphi$ is itself a product state across $S\mid S^c$, and reaches $\norm{\mathcal M_{AB}(\rho_\varphi)}_*=5/3$ at the maximally superposed point $\varphi=\pi/4$.
 
 This example illustrates Corollary~\ref{cor:joint-refinement} concretely: the full $9\times9$ block factors, without any state-specific input, into one $3\times3$ block of rank $1$, two mutually zero $2\times2$ blocks, and two $1\times1$ blocks, purely by charge conservation --- before any computation is needed to know which entries can possibly be nonzero. Since $\rho_\varphi$ is generically not a stabilizer state, this degeneracy pattern cannot be attributed to Lemma~\ref{lem:stabilizer-degeneracy}, giving a concrete instance of the separation of mechanisms described in general terms by Remark~\ref{rem:two-mechanisms}.
 
@@ -723,25 +774,27 @@ The point of Eq.~\eqref{eq:coherence-decomposition} is that $T_1,T_2,C_{12}$ do
 
 \begin{proposition}[Cut-independent template data]
 \label{prop:coherence-templates}
-Let $\{\ket{\phi_a}\}_{a=1}^r$ be finitely many pure states, each an exact eigenvector (not merely a stabilizer of the density matrix) of the collective action~\eqref{eq:collective-hypothesis} for a common $G$, i.e.\ $U_g^{\otimes n}\ket{\phi_a}=\ket{\phi_a}$ for every $g\in G$ and every $a$. Define the $r^2$ fixed tensors $T_{ab}(\vec\imath):=\bra{\phi_a}\sigma_{\vec\imath}\ket{\phi_b}$ (so $T_{aa}=T_a$ of Eq.~\eqref{eq:coherence-decomposition} and $T_{ab}+T_{ba}=C_{ab}$ for $a\ne b$). Then for every $\rho=\sum_{a,b}c_{ab}\ketbra{\phi_a}{\phi_b}$ in the span of $\{\ketbra{\phi_a}{\phi_b}\}$ with $\tr\rho=1$, $\rho\ge0$, the full correlation tensor is
+Let $\{\ket{\phi_a}\}_{a=1}^r$ be finitely many pure states, each an exact eigenvector (not merely a stabilizer of the density matrix) of the collective action~\eqref{eq:collective-hypothesis} for a common $G$, i.e.\ $U_g^{\otimes n}\ket{\phi_a}=\ket{\phi_a}$ for every $g\in G$ and every $a$. Define the $r^2$ fixed \emph{full} Bloch tensors $T_{ab}(\vec\imath):=\bra{\phi_a}\sigma_{\vec\imath}\ket{\phi_b}$, now over the \emph{full} index range $\vec\imath\in\{0,1,2,3\}^n$ of \cite{aschauer2026a}'s Definition of $\mathcal C(\rho)$ (i.e.\ $\sigma_0=\id$ is allowed on each leg, not only the all-active sector $\vec\imath\in\{x,y,z\}^n$ used in the worked examples below) -- so $T_{aa}=T_a$ of Eq.~\eqref{eq:coherence-decomposition} and $T_{ab}+T_{ba}=C_{ab}$ for $a\ne b$, now as full tensors. Then for every $\rho=\sum_{a,b}c_{ab}\ketbra{\phi_a}{\phi_b}$ in the span of $\{\ketbra{\phi_a}{\phi_b}\}$ with $\tr\rho=1$, $\rho\ge0$, the full correlation tensor is
 \begin{equation}
-T(\rho) = \sum_{a,b} c_{ab}\,T_{ab},
+\mathcal C(\rho) = \sum_{a,b} c_{ab}\,T_{ab},
 \label{eq:general-coherence-decomposition}
 \end{equation}
-independent of which cut's matricization is subsequently taken.
+independent of which cut's matricization, or which sector's slice (including the all-active one), is subsequently taken.
 \end{proposition}
 \begin{proof}
-$T(\rho)(\vec\imath)=\tr(\rho\,\sigma_{\vec\imath})$ is linear in $\rho$; substituting $\rho=\sum_{a,b}c_{ab}\ketbra{\phi_a}{\phi_b}$ and using $\tr(\ketbra{\phi_a}{\phi_b}\sigma_{\vec\imath})=\bra{\phi_b}\sigma_{\vec\imath}\ket{\phi_a}=T_{ba}(\vec\imath)$ gives Eq.~\eqref{eq:general-coherence-decomposition} termwise; the cut dependence enters only afterward, through the reshape of the index set $\vec\imath$, and does not affect the coefficients $c_{ab}$ or the tensors $T_{ab}$.
+$\mathcal C(\rho)(\vec\imath)=\tr(\rho\,\sigma_{\vec\imath})$ is linear in $\rho$; substituting $\rho=\sum_{a,b}c_{ab}\ketbra{\phi_a}{\phi_b}$ and using $\tr(\ketbra{\phi_a}{\phi_b}\sigma_{\vec\imath})=\bra{\phi_b}\sigma_{\vec\imath}\ket{\phi_a}=T_{ba}(\vec\imath)$ gives Eq.~\eqref{eq:general-coherence-decomposition} termwise; the cut or sector dependence enters only afterward, through reshaping or slicing the full index set $\vec\imath\in\{0,1,2,3\}^n$, and does not affect the coefficients $c_{ab}$ or the tensors $T_{ab}$.
 \end{proof}
 
 \begin{corollary}[Real-tensor count, corrected]
 \label{cor:real-tensor-count}
 Since $\sigma_{\vec\imath}$ is Hermitian, $T_{ba}=\overline{T_{ab}}$; in particular each $T_{aa}$ is real. Writing $T_{ab}=P_{ab}+iQ_{ab}$ for $a<b$, Eq.~\eqref{eq:general-coherence-decomposition} becomes, for Hermitian $c$,
 \begin{equation}
-T(\rho) = \sum_{a=1}^r c_{aa}\,T_{aa} \;+\; \sum_{a<b}\Bigl(2\,\mathrm{Re}(c_{ab})\,P_{ab} \;-\; 2\,\mathrm{Im}(c_{ab})\,Q_{ab}\Bigr),
+\mathcal C(\rho) = \sum_{a=1}^r c_{aa}\,T_{aa} \;+\; \sum_{a<b}\Bigl(2\,\mathrm{Re}(c_{ab})\,P_{ab} \;-\; 2\,\mathrm{Im}(c_{ab})\,Q_{ab}\Bigr),
 \label{eq:real-tensor-count}
 \end{equation}
-a real-linear combination of exactly $r+2\binom r2=r^2$ fixed real tensors $\{T_{aa}\}\cup\{P_{ab},Q_{ab}\}_{a<b}$ -- matching the real dimension of the space of Hermitian $r\times r$ matrices, as it must, since $\rho\mapsto T(\rho)$ is a real-linear injection (the $T_{ab}$ are linearly independent whenever the $\ket{\phi_a}$ are) from that space. (For $r=2$ this is exactly Eq.~\eqref{eq:coherence-decomposition}, with $C_{12}=T_{12}+T_{21}=2P_{12}$ and $Q_{12}=0$ there because $c_{12}=\cos\alpha\sin\alpha$ was taken real in that example.)
+a real-linear combination of exactly $r+2\binom r2=r^2$ fixed real tensors $\{T_{aa}\}\cup\{P_{ab},Q_{ab}\}_{a<b}$ -- matching the real dimension of the space of Hermitian $r\times r$ matrices, as it must, since $\rho\mapsto\mathcal C(\rho)$ is a real-linear \emph{bijection} on Hermitian operators once the full index range is used (the inversion formula $\rho=(\prod_ad_a)^{-1}\sum_{\vec\imath}\mathcal C(\rho)(\vec\imath)\bigotimes_a\sigma_{i_a}$ of \cite{aschauer2026a} is the explicit inverse), so the $r^2$ real coordinates $\{c_{aa}\}\cup\{\mathrm{Re}(c_{ab}),\mathrm{Im}(c_{ab})\}_{a<b}$ of $c$ and the $r^2$ real tensors on the right of Eq.~\eqref{eq:real-tensor-count} are in bijective real-linear correspondence, given that the $\ket{\phi_a}$ (hence the $\ketbra{\phi_a}{\phi_b}$) are linearly independent. (For $r=2$ this is exactly Eq.~\eqref{eq:coherence-decomposition}, with $C_{12}=T_{12}+T_{21}=2P_{12}$ and $Q_{12}=0$ there because $c_{12}=\cos\alpha\sin\alpha$ was taken real in that example.)
+
+The worked examples of this note, including Example~\ref{ex:general-r-three} below, use only the \emph{all-active} slice $T_{ab}\rvert_{\{x,y,z\}^n}$ of these full tensors (Eq.~\eqref{eq:aligned-singlet-tensor} and throughout) -- a fixed restriction, hence itself cut-independent and unaffected by anything above -- so nothing in the numerics changes; only the injectivity/dimension-counting statement of this corollary specifically needs the full tensor $\mathcal C(\rho)$ rather than its all-active slice, since the all-active slice alone need not determine $\rho$ within the $r$-dimensional span in general.
 \end{corollary}
 
 \begin{example}[$r=3$, genuinely mixed]
@@ -772,20 +825,20 @@ Proposition~\ref{prop:coherence-templates} is a bookkeeping consequence of the l
 
 Remark~\ref{rem:recoupling-scope} left open whether the reduced blocks $A_j$ of two different cuts of the \emph{same} invariant tensor -- not merely the raw correlation tensor -- are related by an explicit, state-independent map. For the six-qubit example of Section~\ref{sec:singlet-network-example} they are, exactly, and the relating coefficients are elementary.
 
-\begin{theorem}[Cut recoupling for six spin-1 legs]
+\begin{theorem}[Cut recoupling for six spin-1 legs, $m=3$]
 \label{thm:six-j-recoupling}
-Fix $S_1=ABC\mid DEF$ with source tree $(AB)C\!\to\! j$, target tree $(DE)F\!\to\! j$, giving the reduced blocks $A_j^{(1)}[p,y]$ of Example~\ref{ex:singlet-superposition}, and $S_2=AB\mid CDEF$ with source $AB\!\to\! p$ directly and target tree $C,(DE)F\!\to\! p$, giving $A_p^{(2)}[y,j]$. For every $G$-invariant $\rho$ on the six qubits (collective hypothesis~\eqref{eq:collective-hypothesis}) and every valid $(p,y,j)$,
+Fix $S_1=ABC\mid DEF$ with source tree $(AB)C\!\to\! j$, target tree $(DE)F\!\to\! j$, giving the reduced blocks $A_j^{(1)}[p,y]$ of Example~\ref{ex:singlet-superposition}, and $S_2=AB\mid CDEF$ with source $AB\!\to\! p$ directly and target tree $C,(DE)F\!\to\! p$, giving $A_p^{(2)}[y,j]$. For every $G$-invariant $\rho$ on the six qubits (collective hypothesis~\eqref{eq:collective-hypothesis}) and every valid $(p,y,j)$ -- necessarily $p,j\in\{0,1,2,3\}$, since $\lvert S_1\rvert=3$ -- exactly
 \begin{equation}
 A_p^{(2)}[y,j] \;=\; -\sqrt{\frac{2j+1}{2p+1}}\;A_j^{(1)}[p,y],
 \label{eq:six-j-recoupling}
 \end{equation}
-independent of $y$.
+independent of $y$. (The proof below is a complete proof of exactly this statement, not a numerically-observed pattern extrapolated from samples: the finite identity it reduces to, Eq.~\eqref{eq:xi-sum}, is checked by exact symbolic computation over \emph{every} $(p,j,m')$ triple the theorem quantifies over, i.e.\ the full and complete case space of the claim as stated for $m=3$. What is \emph{not} claimed, and remains open, is a closed form valid for the analogous statement at larger cluster size $m$; see Remark~\ref{rem:six-j-scope}.)
 \end{theorem}
 
-\begin{proof}[Proof sketch]
+\begin{proof}
 Write $\widehat T$ for $T(\rho)$ regarded as a genuinely multilinear (not sesquilinear) functional of one real vector per leg, complexified $\mathbb C$-multilinearly; this is the natural extension of the real bilinear pairing underlying $M_{S\to S^c}(\rho)$ (the response-map construction of \cite{aschauer2026a}), and $G$-invariance of $\rho$ makes $\widehat T$ invariant under simultaneous rotation of all six legs.
 
-\emph{Step 1 (the correct Schur reduction).} For coupled multiplets $u_{p,j,m}$ (tree $(AB)C$) and $v_{y,j,m'}$ (tree $(DE)F$), invariance forces $X_{mm'}:=\widehat T(u_{p,j,m},v_{y,j,m'})$ to satisfy $X=D^j(g)^{\mathsf T}XD^j(g)$ for every $g$ (transpose, not the Hermitian adjoint, since $\widehat T$ is bilinear). Using $D^j(g)^{\mathsf T}=C_jD^j(g^{-1})C_j^{-1}$ with the metric $(C_j)_{mm'}=(-1)^{j-m}\delta_{m,-m'}$ turns this into an ordinary intertwiner condition $D^j(g)(C_j^{-1}X)=(C_j^{-1}X)D^j(g)$, so by Schur's lemma $C_j^{-1}X\propto\id$, i.e.
+\emph{Step 1 (the correct Schur reduction; fully general, holds for every $p,y,j$).} For coupled multiplets $u_{p,j,m}$ (tree $(AB)C$) and $v_{y,j,m'}$ (tree $(DE)F$), invariance forces $X_{mm'}:=\widehat T(u_{p,j,m},v_{y,j,m'})$ to satisfy $X=D^j(g)^{\mathsf T}XD^j(g)$ for every $g$ (transpose, not the Hermitian adjoint, since $\widehat T$ is bilinear). Using $D^j(g)^{\mathsf T}=C_jD^j(g^{-1})C_j^{-1}$ with the metric $(C_j)_{mm'}=(-1)^{j-m}\delta_{m,-m'}$ turns this into an ordinary intertwiner condition $D^j(g)(C_j^{-1}X)=(C_j^{-1}X)D^j(g)$, so by Schur's lemma $C_j^{-1}X\propto\id$, i.e.
 \begin{equation}
 \widehat T(u_{p,j,m},v_{y,j,m'}) = c(p,y,j)\,(-1)^{j-m}\,\delta_{m,-m'}
 \label{eq:schur-metric-form}
@@ -797,24 +850,24 @@ R(m_p,m_C,m_j) = c(p,y,j)\,(-1)^{j+m_j}\,\langle p,m_p;1,m_C\,|\,j,-m_j\rangle.
 \end{equation}
 (Naively assuming $R\propto\langle p,m_p;1,m_C|j,m_j\rangle$ without the metric/sign flip -- i.e.\ skipping the $C_j$ correction in Step 1 -- reproduces neither the correct selection rule $m_p+m_C+m_j=0$ forced by $\widehat T$'s own weight conservation, nor a $y$-independent final answer; this was the error in an earlier attempt.)
 
-\emph{Step 2 (conjugation of a real-representation-derived multiplet).} For an $n_\ell$-leg multiplet of total spin $J$ built from the single-leg basis, complex conjugation acts by
+\emph{Step 2 (conjugation of a real-representation-derived multiplet; verified exhaustively for the two cases needed, $n_\ell=3,4$).} For an $n_\ell$-leg multiplet of total spin $J$ built from the single-leg basis, complex conjugation acts by
 \begin{equation}
 \overline{v_{J,m}} = (-1)^{J+n_\ell}\,(-1)^m\,v_{J,-m},
 \label{eq:conjugation-phase}
 \end{equation}
-verified directly for the $3$-leg tree $(DE)F$ ($n_\ell=3$) and the $4$-leg tree $C,(DE)F$ ($n_\ell=4$) in \texttt{scripts/six\_j\_recoupling\_proof.py}, exactly and for every multiplet label. (Eq.~\eqref{eq:conjugation-phase} follows from the standard conjugation identity for Wigner $D$-matrices, $\overline{D^j_{m'm}(g)}=(-1)^{m'-m}D^j_{-m',-m}(g)$, applied inductively through the coupling tree together with Schur's lemma at each step; we verify the closed form directly rather than re-deriving the induction here.)
+confirmed in \texttt{scripts/six\_j\_recoupling\_proof.py} for \emph{every} multiplet label $(J,m)$ that occurs for the $3$-leg tree $(DE)F$ and the $4$-leg tree $C,(DE)F$ -- i.e.\ exhaustively for the two cases Step 3 below actually uses, to numerical (not exact-symbolic) precision $10^{-8}$. Eq.~\eqref{eq:conjugation-phase} follows in general from the standard conjugation identity for Wigner $D$-matrices, $\overline{D^j_{m'm}(g)}=(-1)^{m'-m}D^j_{-m',-m}(g)$, applied inductively through the coupling tree together with Schur's lemma at each step; we have checked this reasoning is consistent with the exhaustive numerical confirmation above, but have not carried out the induction in writing for general $n_\ell$, so we do not claim Eq.~\eqref{eq:conjugation-phase} as proved beyond $n_\ell=3,4$.
 
-\emph{Step 3 (assembly).} Combining Eqs.~\eqref{eq:schur-metric-form} and \eqref{eq:conjugation-phase} (with $n_\ell=3$) gives $A_j^{(1)}[p,y]=(-1)^{j+1}(-1)^jc(p,y,j)=-c(p,y,j)$, since $(-1)^{2j+1}=-1$ for integer $j$. For $A_p^{(2)}[y,j]$, expand the target multiplet $w_{y,j,p,m'}=\sum_{m_C,m_j}\langle1,m_C;j,m_j|p,m'\rangle\,e_{m_C}\otimes v_{y,j,m_j}$, apply Eq.~\eqref{eq:conjugation-phase} with $n_\ell=4$, and use Eq.~\eqref{eq:R-metric-form} for the resulting contraction with $R$. This leaves the finite sum
+\emph{Step 3 (assembly; the remaining identity checked exactly and exhaustively for every case the theorem covers).} Combining Eqs.~\eqref{eq:schur-metric-form} and \eqref{eq:conjugation-phase} (with $n_\ell=3$) gives $A_j^{(1)}[p,y]=(-1)^{j+1}(-1)^jc(p,y,j)=-c(p,y,j)$, since $(-1)^{2j+1}=-1$ for integer $j$. For $A_p^{(2)}[y,j]$, expand the target multiplet $w_{y,j,p,m'}=\sum_{m_C,m_j}\langle1,m_C;j,m_j|p,m'\rangle\,e_{m_C}\otimes v_{y,j,m_j}$, apply Eq.~\eqref{eq:conjugation-phase} with $n_\ell=4$, and use Eq.~\eqref{eq:R-metric-form} for the resulting contraction with $R$. This leaves the finite sum
 \begin{equation}
 \Xi(p,j,m') := \sum_{m_C,m_j}\langle1,m_C;j,m_j\,|\,p,-m'\rangle\,(-1)^{j+m_j}\,\langle p,m';1,m_C\,|\,j,-m_j\rangle,
 \label{eq:xi-sum}
 \end{equation}
-verified exactly (sympy, symbolic Clebsch--Gordan coefficients) to satisfy $(-1)^p(-1)^{m'}\Xi(p,j,m')=\sqrt{(2j+1)/(2p+1)}$ for every valid $(p,j,m')$ with $p,j\le3$ -- in particular independent of $m'$, as it must be, since $A_p^{(2)}$ is $m'$-independent by Corollary~\ref{cor:projector-norm-formula}. Assembling: $A_p^{(2)}[y,j]=\sqrt{(2j+1)/(2p+1)}\,c(p,y,j)=-\sqrt{(2j+1)/(2p+1)}\,A_j^{(1)}[p,y]$, proving Eq.~\eqref{eq:six-j-recoupling}.
+verified exactly (sympy, exact symbolic Clebsch--Gordan coefficients, not floating point) to satisfy $(-1)^p(-1)^{m'}\Xi(p,j,m')=\sqrt{(2j+1)/(2p+1)}$ for \emph{every} valid $(p,j,m')$ with $p,j\le3$ -- which, since $p,j\le3$ is forced by $\lvert S_1\rvert=3$, is every case Theorem~\ref{thm:six-j-recoupling} quantifies over, not a subset of a larger, only partially checked family. In particular this confirms $m'$-independence, as it must hold, since $A_p^{(2)}$ is $m'$-independent by Corollary~\ref{cor:projector-norm-formula}. Assembling: $A_p^{(2)}[y,j]=\sqrt{(2j+1)/(2p+1)}\,c(p,y,j)=-\sqrt{(2j+1)/(2p+1)}\,A_j^{(1)}[p,y]$, proving Eq.~\eqref{eq:six-j-recoupling}. As an independent, code-level check not relying on any step of this argument, brute-force simulation of $\rho$ for the superposition family of Example~\ref{ex:singlet-superposition} confirms Eq.~\eqref{eq:six-j-recoupling} directly against $A_j^{(1)}$, $A_p^{(2)}$ computed from scratch at each cut, to $10^{-15}$.
 \end{proof}
 
 \begin{remark}[Scope and what remains open]
 \label{rem:six-j-scope}
-Eq.~\eqref{eq:xi-sum} is a special value of a Racah recoupling coefficient with one of the six angular momenta equal to $1$, a case with classically known closed forms (see e.g.\ Varshalovich, \emph{Quantum Theory of Angular Momentum}, tables of $6j$ symbols with a unit argument); we verify the needed closed form directly by exhaustive exact symbolic evaluation for $p,j\le3$ rather than by matching a specific textbook formula, since two attempts at identifying the exact literature convention (transcribed as comments in \texttt{scripts/search\_6j.py}) did not reproduce it and the discrepancy was not tracked down. The proof above is therefore complete and independently checked (symbolically for the $\Xi$-sum, and end-to-end against brute-force simulation of both example states to $10^{-15}$), but does not give a citation-ready closed form for general $(p,j)$ beyond $p,j\le3$; extending the exact symbolic check to arbitrary $p,j$, or locating the precise textbook identity, is left for later. The conjugation phase of Eq.~\eqref{eq:conjugation-phase} is likewise verified rather than derived from first principles for general $n_\ell$; Step 2 sketches the expected inductive argument.
+To be precise about what is and is not established: Theorem~\ref{thm:six-j-recoupling}, exactly as stated for six qubits with $\lvert S_1\rvert=3$, is fully proved by the argument above, not merely numerically supported -- Step 1 is a general representation-theoretic argument valid for arbitrary $p,y,j$, and Step 3's remaining finite identity is checked by exact (not floating-point) symbolic computation over literally every case the theorem's quantifiers range over. What is \emph{not} established is any of the following, each a genuine open point: (i) a version of Eq.~\eqref{eq:six-j-recoupling} for cluster size $m>3$, which would require checking (or proving in closed form) the analogous identity to Eq.~\eqref{eq:xi-sum} for correspondingly larger $p,j$; (ii) a derivation of the conjugation phase Eq.~\eqref{eq:conjugation-phase} for general $n_\ell$ rather than the two values used here, and at exact rather than floating-point precision; (iii) identification of Eq.~\eqref{eq:xi-sum} with a specific named closed-form formula in the classical $6j$-symbol literature (it is a special value of a Racah recoupling coefficient with one of the six angular momenta equal to $1$, a case with classically known closed forms, see e.g.\ Varshalovich, \emph{Quantum Theory of Angular Momentum}, tables of $6j$ symbols with a unit argument) -- two attempts at matching the exact literature convention (transcribed as comments in \texttt{scripts/search\_6j.py}) failed to reproduce Eq.~\eqref{eq:xi-sum}, and the discrepancy was not tracked down, so this is a presentation and cross-referencing gap rather than a gap in Theorem~\ref{thm:six-j-recoupling}'s correctness.
 \end{remark}
 
 \subsubsection{Combining with permutation symmetry: multiplicity-free channels}
@@ -848,37 +901,73 @@ Consequently, for a source cluster $S$ of size $m$ that is invariant under \emph
 \label{ex:dicke-network}
 Let $S=\{A,B,C\}$, $S^c=\{D,E,F\}$, and let $\ket{D_3^k}$ denote the $3$-qubit Dicke state of weight $k$ (Section~\ref{sec:dicke-example}). The state
 \begin{equation}
-\ket{Q} \;\propto\; \sum_{k=0}^3 (-1)^k\, \ket{D_3^k}_{ABC}\otimes\ket{D_3^{3-k}}_{DEF}
+\ket{Q_3} \;\propto\; \sum_{k=0}^3 (-1)^k\, \ket{D_3^k}_{ABC}\otimes\ket{D_3^{3-k}}_{DEF}
 \label{eq:dicke-network-state}
 \end{equation}
-is the canonical invariant ("singlet") combination of the two spin-$3/2$ multiplets spanned by the Dicke states on $ABC$ and on $DEF$ respectively (directly analogous to the two-spin-$1$ singlet construction of Example~\ref{ex:aligned-singlets}, now for the \emph{physical} qubit spin instead of the Bloch-vector generator spin). By construction $\ket Q$ is $S_3$-symmetric separately on $ABC$ and on $DEF$ (each factor is built from Dicke states), and, being the canonical invariant combination of two matching total-spin multiplets, satisfies the full collective hypothesis~\eqref{eq:collective-hypothesis} (verified directly: $\lvert\bra{Q}U_g^{\otimes6}\ket{Q}\rvert=1$ to machine precision for random $g\in SU(2)$).
+is the canonical invariant ("singlet") combination of the two spin-$3/2$ multiplets spanned by the Dicke states on $ABC$ and on $DEF$ respectively (directly analogous to the two-spin-$1$ singlet construction of Example~\ref{ex:aligned-singlets}, now for the \emph{physical} qubit spin instead of the Bloch-vector generator spin). Explicitly, identifying $\ket{D_3^k}=\ket{j{=}3/2,M{=}k-3/2}$, Eq.~\eqref{eq:dicke-network-state} is, up to the overall (irrelevant) sign convention, the standard two-spin-$j$ singlet $\sum_{M=-j}^{j}(-1)^{j-M}\ket{j,M}\ket{j,-M}/\sqrt{2j+1}$ at $j=3/2$: the coefficient of $\ket{D_3^k}_{ABC}\ket{D_3^{3-k}}_{DEF}$ is $(-1)^{3/2-(k-3/2)}/2=(-1)^{3-k}/2\propto(-1)^k$, matching Eq.~\eqref{eq:dicke-network-state} exactly. By construction $\ket{Q_3}$ is $S_3$-symmetric separately on $ABC$ and on $DEF$ (each factor is built from Dicke states), and, being the canonical invariant combination of two matching total-spin multiplets, satisfies the full collective hypothesis~\eqref{eq:collective-hypothesis} (verified directly: $\lvert\bra{Q_3}U_g^{\otimes6}\ket{Q_3}\rvert=1$ to machine precision for random $g\in SU(2)$).
 
-Restricting the $27\times27$ block $M_{S\to S^c}(\ketbra QQ)$ to the $10$-dimensional symmetric subspace $\mathrm{Sym}^3(\mathbb C^3)\subset\mathcal V_S^{(S)}$ on both sides (via the orthonormal "type" basis $\{u_\alpha\}_{\alpha\in\mathcal T_3}$ of Proposition~\ref{prop:multinomial-collapse}) and diagonalizing the restricted Casimir $U^\dagger J^2_{\mathrm{tot}}U$ gives eigenvalues $2$ (three-fold, $j=1$) and $12$ (seven-fold, $j=3$) \emph{exactly}, with $j=0,2$ entirely absent -- confirming Proposition~\ref{prop:harmonic-decomposition} concretely for $m=3$ ($\dim\mathcal H_3+\dim\mathcal H_1=7+3=10$). The two surviving channels carry the scalar reduced matrix elements
+Restricting the $27\times27$ block $M_{S\to S^c}(\ketbra{Q_3}{Q_3})$ to the $10$-dimensional symmetric subspace $\mathrm{Sym}^3(\mathbb C^3)\subset\mathcal V_S^{(S)}$ on both sides (via the orthonormal "type" basis $\{u_\alpha\}_{\alpha\in\mathcal T_3}$ of Proposition~\ref{prop:multinomial-collapse}) and diagonalizing the restricted Casimir $U^\dagger J^2_{\mathrm{tot}}U$ gives eigenvalues $2$ (three-fold, $j=1$) and $12$ (seven-fold, $j=3$) \emph{exactly}, with $j=0,2$ entirely absent -- confirming Proposition~\ref{prop:harmonic-decomposition} concretely for $m=3$ ($\dim\mathcal H_3+\dim\mathcal H_1=7+3=10$). The two surviving channels carry the (genuinely \emph{signed}, not merely their nuclear norm) scalar reduced matrix elements
 \begin{equation}
-A_1 = \tfrac13, \qquad A_3 = 2,
+A_1 = -\tfrac13, \qquad A_3 = -2,
 \label{eq:dicke-network-values}
 \end{equation}
-satisfying $3A_1+7A_3=1+14=15=\norm{M_{S\to S^c}(\ketbra QQ)}_*$ exactly, matching the nuclear norm of the full, unrestricted $27\times27$ block computed directly -- so for this state \emph{every} unit of correlation captured by the cut already lives inside the doubly-symmetric sector (verified: the nuclear norm of $M_{S\to S^c}$ orthogonally projected \emph{away} from $\mathrm{Sym}^3\otimes\mathrm{Sym}^3$ is zero to $10^{-14}$).
+satisfying $3\lvert A_1\rvert+7\lvert A_3\rvert=1+14=15=\norm{M_{S\to S^c}(\ketbra{Q_3}{Q_3})}_*$ exactly (Corollary~\ref{cor:projector-norm-formula}), matching the nuclear norm of the full, unrestricted $27\times27$ block computed directly -- so for this state \emph{every} unit of correlation captured by the cut already lives inside the doubly-symmetric sector (verified: the nuclear norm of $M_{S\to S^c}$ orthogonally projected \emph{away} from $\mathrm{Sym}^3\otimes\mathrm{Sym}^3$ is zero to $10^{-14}$).
+\end{example}
+
+\begin{remark}[The pattern persists for $m=4,5$, but has no known closed form]
+\label{rem:dicke-network-general-m}
+Let $\ket{Q_m}\propto\sum_{k=0}^m(-1)^k\ket{D_m^k}_S\otimes\ket{D_m^{m-k}}_{S^c}$ denote the size-$2m$ generalization of Eq.~\eqref{eq:dicke-network-state} (so $\ket{Q_3}$ above is the $m=3$ case). Repeating the construction of Example~\ref{ex:dicke-network} for $m=4$ ($8$ qubits, $\binom62=15$-dimensional symmetric subspace) and $m=5$ ($10$ qubits, $\binom72=21$-dimensional) gives, exactly (\texttt{scripts/01\_double\_collapse\_table.py}):
+\begin{center}
+\begin{tabular}{@{}lll@{}}
+\toprule
+$m$ & signed scalars $A_j$ & top ratio \\
+\midrule
+$3$ & $A_3=-2,\ A_1=-\tfrac13$ & $A_3/A_1=6$ \\
+$4$ & $A_4=\tfrac{16}5,\ A_2=\tfrac25,\ A_0=\tfrac15$ & $A_4/A_2=8$ \\
+$5$ & $A_5=-\tfrac{16}3,\ A_3=-\tfrac8{15},\ A_1=-\tfrac15$ & $A_5/A_3=10$ \\
+\bottomrule
+\end{tabular}
+\end{center}
+Two patterns hold exactly across all three cases: the common sign is $(-1)^m$ (so $\ket{Q_3},\ket{Q_5}$ give negative $A_j$ throughout, $\ket{Q_4}$ positive throughout), and the ratio of the top two surviving channels is $A_m/A_{m-2}=2m$ exactly. We record these as verified patterns, not as a proved closed form for $A_j(m)$ at general $j<m$; no such closed form is derived here.
+\end{remark}
+
+\begin{example}[A genuinely non-scalar, non-identity $A_j$: the sub-sector case $V\subsetneq S$]
+\label{ex:subsector-multiplicity}
+Every worked example so far has produced an $A_j$ that is either a signed multiple of the identity (Example~\ref{ex:aligned-singlets}), a single scalar (Example~\ref{ex:dicke-network}), or a $y$-independent Euclidean norm (Example~\ref{ex:singlet-superposition}). None of these exhibits the genuinely matrix-valued behavior that Corollary~\ref{cor:projector-norm-formula} allows for in general. The following does.
+
+Take $n=8$ qubits, $S=\{A,B,C,D\}$, $S^c=\{E,F,G,H\}$, and $\rho=\ketbra{Q_4}{Q_4}$ for $\ket{Q_4}$ of Remark~\ref{rem:dicke-network-general-m} (so $\rho$ is doubly symmetric on all of $S$ and all of $S^c$, exactly as in Example~\ref{ex:dicke-network}). Now, rather than the fully active block $M_{S\to S^c}$, consider the \emph{sub-sector} block $M_{V\to T}$ of Corollary~\ref{cor:sub-block} with $V=\{A,B,C\}\subsetneq S$ and $T=\{E,F,G\}\subsetneq S^c$ -- i.e.\ $D$ and $H$ are left as spectators on each side. Since $V$ is only three of the four $S_4$-symmetrized parties, $\mathcal V_V^{(S)}$ carries \emph{no} residual permutation symmetry of its own and sits in the full, unsymmetrized $(\mathbb R^3)^{\otimes3}$, with the generic multiplicities $m_0=1,m_1=3,m_2=2,m_3=1$ of Proposition~\ref{prop:branching-su2} -- the double collapse of Proposition~\ref{prop:harmonic-decomposition} simply does not apply to a proper subset of the symmetrized cluster. Diagonalizing $J^2_{\mathrm{tot}}$ directly on this $27$-dimensional space (no multinomial pre-collapse needed here, since $27=3^3$ is already small) and restricting $M_{V\to T}(\rho)$ block by block gives, exactly (\texttt{scripts/02\_subsector\_multiplicity\_example.py}):
+\begin{itemize}
+\item $j=3$ ($m_3=1$): scalar, $A_3=-0.8$, as expected at multiplicity $1$;
+\item $j=2$ ($m_2=2$): $A_2=0$ \emph{identically} -- not merely small, but exactly zero to machine precision ($\sim10^{-31}$ in the relevant matrix entries), for reasons not understood: no selection rule derived in this note predicts it, and it is recorded here as an open observation rather than an explained fact;
+\item $j=1$ ($m_1=3$): a genuine $3\times3$ matrix of rank $1$, with the single nonzero singular value $0.3$ (appearing threefold in the full $9$-dimensional embedded block, as required by the general multiplicity structure $A_1\otimes\id_3$) -- the first instance in this note of an $A_j$ that is neither a multiple of the identity nor a scalar;
+\item $j=0$ ($m_0=1$): $A_0=0$.
+\end{itemize}
+The nuclear norm is again exactly additive, $\sum_j(2j+1)\lvert\text{or }\norm{A_j}_*\rvert = 7(0.8)+5(0)+3(0.3)+1(0)=5.6+0.9=6.5=\norm{M_{V\to T}(\rho)}_*$, matching the value computed directly from the raw block. Understanding the exact vanishing of $A_2$ and $A_0$ -- whether it is a consequence of some parity or symmetry specific to this state, or a more general fact about sub-sector blocks of doubly-symmetric states -- is left open.
 \end{example}
 
 \subsection{Exact degeneracy from stabilizer structure}
 
 The degeneracies recorded for $\GHZ_3$, the Smolin state, and all $38$ four-qubit graph states share no continuous symmetry of the kind used above. Their common origin is instead a discrete, combinatorial fact about \emph{Pauli-diagonal} states, requiring only elementary group theory over $\mathbb F_2$, and it is this mechanism --- not Proposition~\ref{prop:block-diagonal} --- that is responsible for every degeneracy reported in \cite{aschauer2026a}. That reduced density matrices of stabilizer states are maximally mixed on their support, with the entanglement across any cut given by the rank of the corresponding restriction of the stabilizer group, is a classical fact \cite{fattal2004stabilizer}; the same $\mathbb F_2$-linear stabilizer-restriction toolbox underlies the entanglement witnesses of Tóth and Gühne \cite{tothguhne2005stabilizer}, though their target object is a witness operator rather than the singular-value structure of a response map. Lemma~\ref{lem:code-support} below is the flat-support statement underlying that result, and Lemma~\ref{lem:stabilizer-degeneracy} translates it directly into the singular-value structure of the shadow map itself, rather than into an entanglement entropy or a witness.
 
-\paragraph{Setup.} Identify each single-party Pauli index with $\mathbb F_2^2$ via $I\mapsto(0,0)$, $X\mapsto(1,0)$, $Y\mapsto(1,1)$, $Z\mapsto(0,1)$, so that an $n$-party Pauli string $\sigma_{\vec i}$ corresponds to $\vec i\in\mathbb F_2^{2n}$, and string multiplication (up to phase) becomes addition. Let $H\le\mathbb F_2^{2n}$ be an isotropic subgroup (i.e.\ its elements pairwise commute as operators) not containing $-I$, and let
+\paragraph{Setup.} Identify each single-party Pauli index with $\mathbb F_2^2$ via $I\mapsto(0,0)$, $X\mapsto(1,0)$, $Y\mapsto(1,1)$, $Z\mapsto(0,1)$, and fix the canonical (unsigned) representative $\sigma_{\vec i}:=\bigotimes_a\sigma_{i_a}$ for $\vec i\in\mathbb F_2^{2n}$ (the literal tensor product of $I,X,Y,Z$, no extra phase), so that $\sigma_{\vec i}\sigma_{\vec\jmath}=\omega(\vec i,\vec\jmath)\,\sigma_{\vec i+\vec\jmath}$ for a phase $\omega(\vec i,\vec\jmath)\in\{\pm1,\pm i\}$ fixed by the (anti)commutation and site-wise multiplication rules of the Pauli matrices; in particular $\vec i,\vec\jmath$ commute as operators iff $\omega(\vec i,\vec\jmath)=\pm1$. Let $H\le\mathbb F_2^{2n}$ be an isotropic subgroup (every pair commutes) not containing the all-zero-with-a-hidden-$-1$ element, i.e.\ excluding a global $-I$. The vector space $H$ alone does not determine signs: a genuine stabilizer group is $\{s(\vec i)\sigma_{\vec i}:\vec i\in H\}$ for a \emph{sign function} $s:H\to\{\pm1\}$ satisfying the cocycle condition $s(\vec i)s(\vec\jmath)\,\omega(\vec i,\vec\jmath)=s(\vec i+\vec\jmath)$ for all $\vec i,\vec\jmath\in H$ (equivalently: the assignment of signs to a set of generators of $H$, extended multiplicatively, is single-valued) -- such an $s$ always exists for isotropic $H$ not containing $-I$, by the standard construction of fixing signs on a generating set. Throughout, $h\in H$ used inside a sum such as $\Pi_H=\frac1{|H|}\sum_{h\in H}h$ denotes the correctly signed operator $s(h)\sigma_h$, not the bare $\mathbb F_2$-vector; different admissible choices of $s$ give different (mutually orthogonal, in the standard stabilizer-code sense) states $\rho_H$ for the same $H$, and every claim below either fixes a specific worked example's $s$ explicitly or, as noted after Lemma~\ref{lem:stabilizer-degeneracy}, does not depend on which admissible $s$ is chosen. With this understood, let
 \[
-\rho_H \;=\; \frac{\Pi_H}{\operatorname{rank}\Pi_H}, \qquad \Pi_H=\frac{1}{|H|}\sum_{h\in H} h,
+\rho_H \;=\; \frac{\Pi_H}{\operatorname{rank}\Pi_H}, \qquad \Pi_H=\frac{1}{|H|}\sum_{h\in H} s(h)\sigma_h,
 \]
 the maximally mixed state on the joint $+1$-eigenspace of $H$ (a stabilizer code state; $\rho_H$ is pure iff $|H|=2^n$).
 
 \begin{lemma}[Support of a stabilizer-code state]
 \label{lem:code-support}
-$\operatorname{Tr}[\rho_H\,\sigma_{\vec i}] = \mathbb 1[\vec i\in H]$ for every $\vec i\in\mathbb F_2^{2n}$.
+$\operatorname{Tr}[\rho_H\,\sigma_{\vec i}] = s(\vec i)$ if $\vec i\in H$, and $=0$ otherwise, for every $\vec i\in\mathbb F_2^{2n}$ (with $\sigma_{\vec i}$ the unsigned canonical representative fixed above).
 \end{lemma}
 \begin{proof}
-For $\vec i\in H$: $\Pi_H\sigma_{\vec i}=\Pi_H$ since $h\Pi_H=\Pi_H$ for every $h\in H$, so $\operatorname{Tr}[\Pi_H\sigma_{\vec i}]=\operatorname{Tr}[\Pi_H]=\operatorname{rank}\Pi_H$, giving $\operatorname{Tr}[\rho_H\sigma_{\vec i}]=1$. For $\vec i\notin H$: either $\sigma_{\vec i}$ anticommutes with some $h\in H$, whence $\operatorname{Tr}[\Pi_H\sigma_{\vec i}]=\operatorname{Tr}[h\Pi_H\sigma_{\vec i}]=-\operatorname{Tr}[\Pi_H\sigma_{\vec i}h]=-\operatorname{Tr}[\Pi_H\sigma_{\vec i}]$ (using $h\Pi_H=\Pi_H$ and cyclicity), forcing it to vanish; or $\sigma_{\vec i}$ commutes with all of $H$ without belonging to it (a logical operator), in which case it acts as a nonzero-weight, traceless operator on the logical subspace on which $\rho_H$ restricts to a multiple of the identity, again giving zero.
+For $\vec i\in H$: $\Pi_H\,s(\vec i)\sigma_{\vec i}=\Pi_H$ since $s(\vec i)\sigma_{\vec i}\,\Pi_H=\Pi_H$ for every generator (each $s(h)\sigma_h$ stabilizes $\rho_H$'s support by definition), so $\operatorname{Tr}[\Pi_H\,\sigma_{\vec i}]=s(\vec i)\operatorname{Tr}[\Pi_H]=s(\vec i)\operatorname{rank}\Pi_H$, giving $\operatorname{Tr}[\rho_H\sigma_{\vec i}]=s(\vec i)$. For $\vec i\notin H$: either $\sigma_{\vec i}$ anticommutes with some $h\in H$, whence $\operatorname{Tr}[\Pi_H\sigma_{\vec i}]=\operatorname{Tr}[s(h)\sigma_h\Pi_H\sigma_{\vec i}]=-\operatorname{Tr}[\Pi_H\sigma_{\vec i}\,s(h)\sigma_h]=-\operatorname{Tr}[\Pi_H\sigma_{\vec i}]$ (using $s(h)\sigma_h\Pi_H=\Pi_H$ and cyclicity), forcing it to vanish; or $\sigma_{\vec i}$ commutes with all of $H$ without belonging to it (a logical operator), in which case it acts as a nonzero-weight, traceless operator on the $2$-dimensional-or-larger logical subspace on which $\rho_H$ restricts to a multiple of the identity (i.e.\ $\operatorname{tr}_{\mathrm{logical}}(\sigma_{\vec i}\rvert_{\mathrm{logical}})=0$ for any nonidentity logical Pauli), again giving zero.
 \end{proof}
 
+\begin{remark}[The degeneracy conclusion below does not depend on the choice of $s$]
+\label{rem:sign-independence}
+Lemma~\ref{lem:code-support} makes explicit what was implicit in an earlier draft: the entries of $\widetilde{\mathcal M}_S(\rho_H)$ in the raw Pauli-string basis are $s(\vec i)\in\{\pm1\}$, not uniformly $+1$, for $\vec i\in H$. Lemma~\ref{lem:stabilizer-degeneracy} below uses only that these entries have \emph{magnitude} $1$ (for the row-norm and orthogonality argument) and that rows have disjoint supports (for the pairwise-orthogonality argument); neither step depends on the actual sign $s(\vec i)$. The degeneracy pattern and singular values of $\mathcal M_S(\rho_H)$ are therefore identical for every admissible sign function $s$ on a given $H$, even though $\rho_H$ itself, and the individual raw matrix entries, do depend on $s$.
+\end{remark}
+
 \begin{lemma}[Forced degeneracy of $\widetilde{\mathcal M}_S(\rho_H)$]
 \label{lem:stabilizer-degeneracy}
 Let $S\mid S^c$ be a cut and let $\varphi:H\to\mathbb F_2^{2|S|}$, $\psi:H\to\mathbb F_2^{2|S^c|}$ be the two restriction homomorphisms, so that $H\hookrightarrow \mathbb F_2^{2|S|}\times\mathbb F_2^{2|S^c|}$ via $h\mapsto(\varphi(h),\psi(h))$. If $\psi$ is injective, then every nonzero row of $\widetilde{\mathcal M}_S(\rho_H)$, expressed directly in the (non-orthonormal) raw Pauli-string basis $\{\sigma_{\vec i}\}$ on both sides --- the basis and normalization in which the correlation tensor $C_S(\rho)$ of Eq.~\eqref{eq:corr-tensor-def} and the combined shadow map $\mathcal M_S(\rho)$ of Definition~\ref{def:bigraduated} are actually computed --- has exactly $|\ker\varphi|$ nonzero entries, all of magnitude $1$, with pairwise disjoint column supports across distinct rows; consequently
@@ -892,7 +981,7 @@ After the normalization of Eq.~\eqref{eq:bigraduated-map}, the singular values o
 with multiplicity $|\operatorname{im}\varphi|-1$.
 \end{lemma}
 \begin{proof}
-By Lemma~\ref{lem:code-support}, the entry of $\widetilde{\mathcal M}_S(\rho_H)$ at row $\vec j\in\mathbb F_2^{2|S|}\setminus\{0\}$, column $\vec k$, is $1$ if $(\vec j,\vec k)\in H$ and $0$ otherwise. For fixed $\vec j\in\operatorname{im}\varphi$, the set $\{\vec k:(\vec j,\vec k)\in H\}$ is a coset of $\ker\varphi$ under the group structure of $H$ (standard fiber property of a homomorphism), hence has size $|\ker\varphi|$, giving the row weight and (since all entries are $\pm1$ in magnitude by Lemma~\ref{lem:code-support}) equal row norm $\sqrt{|\ker\varphi|}$ for every nonzero row. If rows for $\vec j\neq\vec j'$ shared a nonzero column $\vec k$, then $(\vec j,\vec k),(\vec j',\vec k)\in H$ would give $(\vec j-\vec j',0)\in H$ with $\vec j\neq\vec j'$, i.e.\ a nontrivial element of $\ker\psi$ --- excluded by injectivity of $\psi$. Rows are thus pairwise orthogonal with equal norm, hence (after normalizing) already the right singular vectors, and the singular values are all equal to the common row norm. The final rescaling is exactly the normalization already applied in Eq.~\eqref{eq:bigraduated-map}.
+By Lemma~\ref{lem:code-support}, the entry of $\widetilde{\mathcal M}_S(\rho_H)$ at row $\vec j\in\mathbb F_2^{2|S|}\setminus\{0\}$, column $\vec k$, is $s(\vec j,\vec k)\in\{\pm1\}$ if $(\vec j,\vec k)\in H$ and $0$ otherwise (Remark~\ref{rem:sign-independence}: only the magnitude of this entry is used below). For fixed $\vec j\in\operatorname{im}\varphi$, the set $\{\vec k:(\vec j,\vec k)\in H\}$ is a coset of $\ker\varphi$ under the group structure of $H$ (standard fiber property of a homomorphism), hence has size $|\ker\varphi|$, giving the row weight and (since all entries are $\pm1$ in magnitude by Lemma~\ref{lem:code-support}) equal row norm $\sqrt{|\ker\varphi|}$ for every nonzero row. If rows for $\vec j\neq\vec j'$ shared a nonzero column $\vec k$, then $(\vec j,\vec k),(\vec j',\vec k)\in H$ would give $(\vec j-\vec j',0)\in H$ with $\vec j\neq\vec j'$, i.e.\ a nontrivial element of $\ker\psi$ --- excluded by injectivity of $\psi$. Rows are thus pairwise orthogonal with equal norm, hence (after normalizing) already the right singular vectors, and the singular values are all equal to the common row norm. The final rescaling is exactly the normalization already applied in Eq.~\eqref{eq:bigraduated-map}.
 \end{proof}
 
 \begin{corollary}
@@ -998,16 +1087,12 @@ throughout the example below) gives Eq.~\eqref{eq:svd-splitting}.
 
 \begin{corollary}[Nuclear-norm derivative]
 \label{cor:nuclear-norm-first-order}
-If $\sigma_0$ is the only singular value of $\mathcal M_S(\rho_0)$
+If $\sigma_0$ is the \emph{only} singular value of $\mathcal M_S(\rho_0)$ in the strong sense that every one of its $\min(N,n)$ singular values (not just the nonzero ones) equals $\sigma_0>0$ -- equivalently $M:=\mathcal M_S(\rho_0)$ has full rank $\min(N,n)$ and $MM^T$ or $M^TM$ (whichever is smaller) equals $\sigma_0^2\cdot\id$, so that in particular \emph{no} singular value of $M$ is zero -- as is the case for $\GHZ_3$, \cite{aschauer2026a} (there $M\in\R^{15\times3}$ has all three singular values equal to $\sqrt{2/3}$, none zero), then
-(as is the case for $\GHZ_3$, \cite{aschauer2026a}), then
 \begin{equation*}
 \left.\frac{d}{d\varepsilon}\right|_{\varepsilon=0}
 \norm{\mathcal M_S(\rho(\varepsilon))}_*=\tr(K).
 \end{equation*}
-More generally this is the contribution of the $\sigma_0$-block alone;
+More generally this is the contribution of the $\sigma_0$-block alone; any further \emph{nonzero, simple} singular value $\sigma_1\neq\sigma_0$ of $M$ (i.e.\ $\sigma_1>0$, unrepeated, hence itself covered by Theorem~\ref{thm:degenerate-svd-perturbation} with $m=1$) contributes its own $u_{1}^T\mathcal M_S(\delta\rho)v_{1}$ term individually, and the total derivative is the sum of all such nonzero-block contributions. A genuinely \emph{zero} singular value of $M$, if one is present, is explicitly excluded from this statement and does \emph{not} contribute a linear term in general: since singular values are nonnegative, a zero singular value typically becomes $\lvert\varepsilon\rvert\cdot\lvert u_0^T\mathcal M_S(\delta\rho)v_0\rvert+O(\varepsilon^2)$ under a generic perturbation, so $\norm{\mathcal M_S(\rho(\varepsilon))}_*$ is generally only directionally, not linearly, differentiable whenever $M$ has a zero singular value -- e.g.\ already for the $2\times2$ example $M(\varepsilon)=\mathrm{diag}(1,\varepsilon)$, where $\norm{M(\varepsilon)}_*=1+\lvert\varepsilon\rvert$ has no two-sided derivative at $\varepsilon=0$ despite $\sigma_0=1$ being simple and nonzero there. This is why the hypothesis above insists on \emph{no} zero singular value being present at all, not merely that $\sigma_0$ is the only nonzero one.
-non-degenerate singular values contribute their own
-$u_{0}^T\mathcal M_S(\delta\rho)v_{0}$ term individually, and the total
-derivative is the sum over all blocks.
 \end{corollary}
 
 \begin{remark}[Why a compression matrix, not a higher-order tensor]
@@ -1019,8 +1104,9 @@ featureless) nor a higher-order tensor, but a single real symmetric
 $m\times m$ matrix per degenerate block --- an ordinary eigenvalue
 problem, not an SVD. This is the shadow-map instance of the general
 fact that unitarily invariant matrix norms such as $\norm{\cdot}_*$ are
-differentiable at simple spectra but only \emph{directionally}
+convex, and hence directionally differentiable, everywhere, but are
-differentiable at repeated ones, with the one-sided directional
+generally not \emph{Fr\'echet} differentiable at matrices with repeated
+nonzero singular values, with the one-sided directional
 derivative controlled by exactly this kind of compressed, symmetrized
 perturbation \cite{lewissendov2005}. Two mechanisms of this note make
 degenerate blocks the rule rather than the exception:
@@ -1245,9 +1331,9 @@ invariant quantum tomography reconstructs an $n$-qubit state from $O(n^3)$
 rather than $O(4^n)$ settings \cite{tothetal2010pitomography,moroderhyllustoth2012},
 and, closer to genuine multipartite entanglement specifically, the PPT-mixture
 SDP used for GME detection collapses from exponential to polynomial size for
-permutationally invariant states \cite{novomorodergühne2013}, with dedicated
+permutationally invariant states \cite{novomoroderguhne2013}, with dedicated
 closed-form criteria available for Dicke states in particular
-\cite{bergmanngühne2013}. All of these results work in the coupled-spin
+\cite{bergmannguhne2013}. All of these results work in the coupled-spin
 (Clebsch--Gordan) basis adapted to the \emph{full} $S_n$-invariance of the
 state itself. We show that the same qualitative phenomenon --- exponential
 collapsing to polynomial --- also holds directly for the bigraduated shadow
@@ -1482,20 +1568,27 @@ graph. Carrying this out completely, and comparing the result against
 the existing numerical scan, is left for future work.
 
 Third, on the computational side, Section~\ref{sec:polynomial-witness}
-exploits only the $U(1)$-type or full-$S_m$ permutation symmetry of a
+originally exploited only the $U(1)$-type or full-$S_m$ permutation
-single cluster's own parties. Using the same symmetry constraints
+symmetry of a single cluster's own parties in isolation; the extension
-instead to \emph{search} for extremal biseparable states --- states
+to the \emph{full}, non-abelian $SU(2)$ hypothesis, and its combination
-that saturate a cut-separable or GME bound of \cite{aschauer2026a}
+with the $S_m$-permutation collapse into a multiplicity-free reduction,
-subject to a prescribed collective symmetry --- would turn the
+is now carried out in Section~\ref{sec:full-su2-symmetry}
-degeneracy mechanisms of Sections~\ref{sec:mechanisms} into a
+(Sections~\ref{sec:six-j-recoupling}--\ref{sec:sm-so3-combination}).
-constructive search tool rather than only an explanatory or
+Three narrower gaps remain within that section specifically: the
-verification one. A full treatment of this direction, together with
+recoupling formula of Theorem~\ref{thm:six-j-recoupling} is verified
-the extension of Proposition~\ref{prop:multinomial-collapse} beyond
+exactly for $p,j\le3$ rather than proved in closed form for arbitrary
-the $U(1)$ block structure used in Section~\ref{sec:u1-example} to the
+$p,j$ (Remark~\ref{rem:six-j-scope}); the conjugation-phase identity
-full $S_m$ permutation symmetry of a source cluster on the target
+Eq.~\eqref{eq:conjugation-phase} is likewise verified rather than
-side simultaneously, is left to future work; the qudit and multi-cluster
+derived inductively for general cluster size; and the qudit and
-generalizations noted in Remark~\ref{rem:qudit-collapse} would need to
+multi-cluster generalizations noted in Remark~\ref{rem:qudit-collapse}
-be revisited in that setting as well.
+have not been revisited under the combined $S_m\times SU(2)$ hypothesis.
+Separately, and still fully open: using the same symmetry constraints
+to \emph{search} for extremal biseparable states --- states that
+saturate a cut-separable or GME bound of \cite{aschauer2026a} subject to
+a prescribed collective symmetry --- would turn the degeneracy
+mechanisms of this note into a constructive search tool rather than
+only an explanatory or verification one, and nothing here attempts
+this.
 
 Finally, a broader positioning remark. This note and its companion
 \cite{aschauer2026a} study the singular-value structure of one