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 two results below need genuinely different amounts of symmetry, and it is worth separating them explicitly rather than proving both under the stronger hypothesis throughout.
\emph{Weak (cut-factorizing) hypothesis.} Let $G$ be a compact group acting on the system by unitaries of the form $g\mapsto U_g^{(S)}\otimes U_g^{(S^c)}$ for some fixed bipartition $S\mid S^c$ -- i.e.\ $G$ acts unitarily on each side of the cut, with no further requirement on how $U_g^{(S)}$ itself decomposes over the parties inside $S$ -- and suppose $\rho$ is $G$-invariant. This induces adjoint representations $\mathrm{Ad}^{(S)}_g$ on $\V_0^{(S)}$ and $\mathrm{Ad}^{(S^c)}_g$ on $\V_0^{(S^c)}$, with isotypic decompositions
where $\lambda$ ranges over the irreducible representations of $G$ appearing on either side, $V_\lambda$ denotes a fixed model of the irreducible representation of dimension $d_\lambda$, and $M_\lambda^{(S)}$, $M_\lambda^{(S^c)}$ are the corresponding multiplicity spaces. This is exactly the same type of decomposition already used for the source and target sector gradings in \cite{aschauer2026a}, now taken with respect to a representation-theoretic rather than a combinatorial grading; the two coincide only in special cases.
This weak hypothesis is exactly what Proposition~\ref{prop:block-diagonal} below uses: its proof only invokes equivariance of $\widetilde{\mathcal M}_S(\rho)$ between the two representations $\mathrm{Ad}^{(S)}_g$, $\mathrm{Ad}^{(S^c)}_g$, and never asks how $U_g^{(S)}$ or $U_g^{(S^c)}$ further factor over the parties within $S$ or within $S^c$.
\emph{Strong (party-local) hypothesis.} Proposition~\ref{prop:sector-symmetry-compatible} and Corollary~\ref{cor:joint-refinement} need more: there $G$ must act by unitaries of the form $g\mapsto\bigotimes_{a\in P}U_g^{(a)}$, one factor per party, since the sector grading $V\subseteq S$ that these results are compatible with is itself defined party by party (Eq.~\eqref{eq:source-bloch-decomposition}). Every party-local action is in particular cut-factorizing for any bipartition, so the strong hypothesis is a special case of the weak one; the converse fails, as the remark following Corollary~\ref{cor:joint-refinement} makes concrete.
The general strategy --- twirl a state over a symmetry group and use Schur's lemma to collapse the resulting computation onto the much smaller multiplicity spaces --- is the same one used classically to simplify the computation of entanglement measures for symmetric states \cite{vollbrechtwerner2001}; the present section applies it to the shadow map itself rather than to a scalar entanglement measure.
\begin{proposition}[Exact block-diagonality]
\label{prop:block-diagonal}
Under the weak, cut-factorizing hypothesis above, $\widetilde{\mathcal M}_S(\rho)$ is block diagonal with respect to the isotypic decomposition~\eqref{eq:isotypic-decomp}: writing $\Pi_\lambda^{(S^c)}$, $\Pi_\mu^{(S)}$ for the isotypic projections,
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
$G$-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.
\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}.
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]
\label{prop:sector-symmetry-compatible}
Let $G$ act on the system under the strong, party-local hypothesis, $g\mapsto\bigotimes_{a\in P}U_g^{(a)}$, and let $S\subseteq P$. Writing $R(g):=\bigotimes_{a\in S}\mathrm{Ad}_{U_g^{(a)}}$ for the induced action on $\V^{(S)}$, we have
\begin{equation}
R(g)\,\V_V^{(S)} = \V_V^{(S)}
\qquad\text{for every } g\in G \text{ and every nonempty } V\subseteq S.
\label{eq:sector-invariance}
\end{equation}
Consequently the sector projectors $P_V$ of Eq.~\eqref{eq:source-bloch-decomposition} commute with $R(g)$ for every $g\in G$, hence with every isotypic projector $P_\lambda=d_\lambda\int_G\chi_\lambda(g)^*R(g)\,dg$ of Eq.~\eqref{eq:isotypic-decomp}, and $\V^{(S)}$ decomposes simultaneously as
Since $\mathrm{Ad}_{U_g^{(a)}}$ fixes the identity, $\mathrm{Ad}_{U_g^{(a)}}\id^{(a)}=\id^{(a)}$, and maps the traceless Hermitian subspace $\V_0^{(a)}$ to itself (conjugation by a unitary preserves both Hermiticity and tracelessness), each local factor space splits $G$-invariantly as $\mathbb C\,\id^{(a)}\oplus\V_0^{(a)}$. The sector space $\V_V^{(S)}$ is, by definition, exactly the span of product basis elements with the $a$-th factor in $\V_0^{(a)}$ for $a\in V$ and equal to $\id^{(a)}$ for $a\notin V$. Since $R(g)$ acts factor-wise and each local factor preserves its own identity-versus-traceless split, $R(g)$ cannot move an element of $\V_V^{(S)}$ out of $\V_V^{(S)}$: it maps trivial legs to trivial legs and active legs to (possibly rotated, but still traceless) active legs, without ever changing which legs are trivial. This gives $R(g)\V_V^{(S)}\subseteq\V_V^{(S)}$; the same argument applied to $g^{-1}$ gives the reverse inclusion, hence Eq.~\eqref{eq:sector-invariance}. Commutation of $P_V$ with $R(g)$, and therefore with the integral defining $P_\lambda$, follows immediately.
\end{proof}
\begin{corollary}[Joint refinement of the bigraduated shadow map]
\label{cor:joint-refinement}
Let $S\mid S^c$ be a cut preserved by $G$ as a set partition, with $G$ acting locally by conjugation on both $S$ and $S^c$ as above. Then, whenever $\rho$ is $G$-invariant, every bigraduated block $M_{V\to T}(\rho)$ of Definition~\ref{def:bigraduated} further decomposes as
for reduced maps $A_{V,T,\lambda}$ on the corresponding multiplicity spaces. In particular, the combinatorial sector grading of \cite{aschauer2026a} and the representation-theoretic isotypic grading of the present section are not competing organizations of $\mathcal M_S(\rho)$, but two orthogonal refinements of the same operator, and can be applied jointly: one may first restrict to a sub-block witness $M_{V\to T}$ as in Corollary~\ref{cor:sub-block} and then further resolve it by symmetry channel, or apply the two refinements in the opposite order, with the same result.
\end{corollary}
\begin{proof}
By Proposition~\ref{prop:sector-symmetry-compatible} applied on the source side to $\V_V^{(S)}$ and, symmetrically, on the target side to $\V_T^{(S^c)}$, the restrictions $R_S(g)\big|_{\V_V^{(S)}}$ and $R_{S^c}(g)\big|_{\V_T^{(S^c)}}$ are themselves well-defined representations of $G$. Since $\rho$ is $G$-invariant, $\widetilde{\mathcal M}_S(\rho)$ intertwines $R_S(g)$ and $R_{S^c}(g)$ on the full spaces (Proposition~\ref{prop:block-diagonal}), and since the sector inclusion $\iota_V$ and sector projection $P_T$ used in Definition~\ref{def:bigraduated} are themselves $G$-equivariant by Proposition~\ref{prop:sector-symmetry-compatible}, the composite $M_{V\to T}(\rho)=P_T\,\widetilde{\mathcal M}_S(\rho)\,\iota_V$ intertwines the restricted representations on $\V_V^{(S)}$ and $\V_T^{(S^c)}$. Schur's lemma applied to this restricted intertwiner gives Eq.~\eqref{eq:joint-block}.
\end{proof}
\begin{remark}[Party permutation: an example separating the two hypotheses]
\label{rem:permutation-separating-example}
Let $\pi\in S_n$ be a permutation of the parties fixing $S$ setwise (i.e.\ $\pi(S)=S$, hence also $\pi(S^c)=S^c$), and let $U_\pi$ be the corresponding permutation unitary on $\mathcal H^{(P)}$. If $\rho$ is invariant under $U_\pi$, this is a cut-factorizing symmetry -- it acts as some unitary on $\mathcal H^{(S)}$ tensored with some unitary on $\mathcal H^{(S^c)}$ -- so Proposition~\ref{prop:block-diagonal} applies unchanged and $\widetilde{\mathcal M}_S(\rho)$ block-diagonalizes over the isotypic components of the induced permutation representations on $\V_0^{(S)}$, $\V_0^{(S^c)}$. But $U_\pi$ is not party-local whenever $\pi$ moves two or more parties within $S$: it acts as a genuine swap between the factors of $\mathcal H^{(S)}$, not as $\bigotimes_{a\in S}U_g^{(a)}$ for a single fixed single-party unitary $U_g$. Consequently Proposition~\ref{prop:sector-symmetry-compatible} does not apply, and indeed its conclusion can fail: $U_\pi$ can map a sector $\V_V^{(S)}$ to a different sector $\V_{\pi(V)}^{(S)}$ rather than fixing it, so the combinatorial sector grading and the permutation-isotypic grading are, in general, genuinely different, incompatible refinements of $\V_0^{(S)}$ -- unlike the party-local case of Corollary~\ref{cor:joint-refinement}, where the two gradings were shown to be compatible refinements of one another.
Two families used elsewhere in this note calibrate this gap. The state $W_n$ is invariant under the full party-permutation group $S_n$, and permutation-invariant under any $\pi$ fixing any given $S$ setwise; it therefore block-diagonalizes under Proposition~\ref{prop:block-diagonal} for every cut, yet it is not a stabilizer state (its single-party marginal has eigenvalues $1/n,(n-1)/n$, never the flat spectrum that Lemma~\ref{lem:code-support} forces on a stabilizer-code reduction), so its degeneracies are explained by permutation symmetry alone, with neither Proposition~\ref{prop:sector-symmetry-compatible} nor Lemma~\ref{lem:stabilizer-degeneracy} available. The ring graph state, by contrast, is invariant only under the cyclic subgroup $C_n\le S_n$, which for $n=4$ and $S=\{1,2\}$ already fails to fix $S$ setwise (the generating shift sends $\{1,2\}\mapsto\{2,3\}$), so Proposition~\ref{prop:block-diagonal} does not apply to that cut at all in the permutation channel; its observed degeneracies, as Corollary~\ref{cor:stabilizer-examples} already notes, are accounted for entirely by the stabilizer mechanism instead. Party-permutation symmetry is thus a third selection mechanism, logically independent of both the collective-unitary mechanism of Proposition~\ref{prop:block-diagonal} under the strong hypothesis and the stabilizer mechanism of Lemma~\ref{lem:stabilizer-degeneracy}, and satisfying it need not entail satisfying either of the other two.
\end{remark}
\begin{example}[Ring graph state: reflections, not rotations, supply the example]
\label{ex:ring-reflection}
The cyclic group $C_4=\langle(1234)\rangle$ stabilizing the ring graph state $\rho_\square$ of \cite{aschauer2026a} indeed fixes no nontrivial cut setwise, as anticipated above. But $\mathrm{Aut}(\square)$ is the full dihedral group $D_4$, of order $8$, and its reflections do fix cuts setwise. For $S=\{1,2\}$, $S^c=\{3,4\}$ (the ``adjacent'' cut of \cite{aschauer2026a}), the reflection $\pi=(12)(34)$ is a graph automorphism with $\pi(S)=S$, $\pi(S^c)=S^c$; it is a genuine two-party swap on each side of the cut, hence exactly the situation of Remark~\ref{rem:permutation-separating-example}: the weak hypothesis holds for $\rho_\square$ at this cut, the strong one does not, since swapping qubits $1$ and $2$ is not of the form $U^{(1)}\otimes U^{(2)}$.
Concretely, $R(\pi)$ does not fix $\V_{\{1\}}^{(S)}$: it maps it onto $\V_{\{2\}}^{(S)}$, since $\pi$ relabels qubit $1$ as qubit $2$. So Proposition~\ref{prop:sector-symmetry-compatible}'s hypothesis fails non-vacuously here, exactly as claimed. What Proposition~\ref{prop:block-diagonal} (the weak statement) still guarantees is equivariance of the full block $\widetilde{\mathcal M}_S(\rho_\square)$ under the combined swap on $\V_{\{1\}}^{(S)}\oplus\V_{\{2\}}^{(S)}$ and $\V_{\{3\}}^{(S^c)}\oplus\V_{\{4\}}^{(S^c)}$; restricted to the four singleton-to-singleton blocks this is the concrete prediction
i.e.\ that the $2\times2$ array of blocks is circulant --- a prediction that follows from equivariance alone, before any stabilizer computation is performed.
A direct check against the stabilizer group $H=\langle K_1,K_2,K_3,K_4\rangle$ of $\rho_\square$ (Lemma~\ref{lem:code-support}) confirms this: in the raw Pauli-string basis, $X_1X_3\in H$ and $X_2X_4\in H$, while $X_1X_4\notin H$ and $X_2X_3\notin H$, so
\begin{equation*}
M_{\{1\}\to\{3\}}(\rho_\square)=M_{\{2\}\to\{4\}}(\rho_\square)=1\ (\text{support on the }xx\text{ direction only}),
matching Eq.~\eqref{eq:ring-circulant-prediction} exactly. This is a case where the weak, cut-factorizing symmetry and the combinatorial stabilizer mechanism of Lemma~\ref{lem:stabilizer-degeneracy} operate on the same numbers from two independent directions: the stabilizer mechanism explains \emph{why} the entries are $0$ or $\pm1$ at all, while the swap-equivariance of Proposition~\ref{prop:block-diagonal} explains, without reference to the stabilizer formalism, why the two nonzero entries must coincide and the two zero entries must coincide. Diagonalizing this $2\times2$ circulant block in the symmetric/antisymmetric basis $(\,e_x^{(1)}\pm e_x^{(2)})/\sqrt2$ gives eigenvalues $1\pm0=1$ in this instance --- here the two channels happen to be degenerate, since the off-diagonal entry vanishes, but the circulant form Eq.~\eqref{eq:ring-circulant-prediction} would hold with generically distinct symmetric/antisymmetric eigenvalues $a\pm b$ for any $\rho$ merely invariant under this same reflection, stabilizer or not.
The ``diagonal'' cut $S=\{1,3\}$, $S^c=\{2,4\}$ of \cite{aschauer2026a} admits the analogous reflection $\pi'=(13)$ (fixing qubits $2,4$ individually and swapping $1,3$ within $S$), giving the same story with $S$ and $S^c$ playing asymmetric roles: here $\pi'$ swaps only within $S$, not within $S^c$, so the weak hypothesis still applies, but now with $R_{S^c}(\pi')$ trivial and $R_S(\pi')$ a nontrivial swap. Equivariance then forces the weaker, one-sided prediction $M_{\{1\}\to T}(\rho_\square)=M_{\{3\}\to T}(\rho_\square)$ for every $T\subseteq\{2,4\}$, again checkable directly against $H$.
\end{example}
\begin{lemma}[Transitive party symmetry forces cut-value equality]
\label{lem:transitive-symmetry}
Suppose all local dimensions coincide, $d_a=d$ for every $a\in P$, and let $H\le S_n$ act on $P$ by permuting parties via the induced permutation unitaries $U_\pi$ on $\mathcal H^{(P)}=(\mathcal H^{(1)})^{\otimes n}$. Suppose $\rho$ is $H$-invariant, $U_\pi\rho U_\pi^\dagger=\rho$ for every $\pi\in H$. Then for every $a\in P$ and $\pi\in H$,
for any single choice of $a\in P$ -- in particular, computing one shadow map suffices instead of all $n$.
\end{lemma}
\begin{proof}
Fix $\pi\in H$ and $a\in P$. Relabeling parties by $\pi$ is a relabeling of tensor factors, so $\mathcal M_{\pi(a)}(U_\pi\rho U_\pi^\dagger)$ is, by construction, exactly the same operator as $\mathcal M_a(\rho)$ transported through the canonical identification of $\mathcal H^{(a)}$ with $\mathcal H^{(\pi(a))}$ (which exists since $d_a=d$ for all $a$); in particular it has the same singular values, so
By hypothesis $U_\pi\rho U_\pi^\dagger=\rho$, giving Eq.~\eqref{eq:transitive-equal-norms}. If $H$ is transitive, the orbit of any fixed $a$ under $H$ is all of $P$, so $\norm{\mathcal M_b(\rho)}_*$ takes the same value for every $b\in P$; Eq.~\eqref{eq:transitive-collapse} is then immediate from the definitions of $\Phi_{\mathrm{sym}}$, $\Phi_{\max}$.
\end{proof}
\subsection{Exact formula and its consequence}
\begin{proposition}[Reduced matrix element formula]
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
where $\tilde r_\lambda$, $\tilde v_\lambda$ are $r_\lambda$, $v_\lambda$ reshaped as $(\dim M_\lambda^{(S)})\times d_\lambda$ and $(\dim M_\lambda^{(S^c)})\times d_\lambda$ matrices in a basis of $V_\lambda$ shared by both sides. Consequently,
For fixed $g$, the product term $(\mathrm{Ad}^{(S)}_g r^{(S)})(\mathrm{Ad}^{(S^c)}_g v_{S^c})^T$ (unnormalized) is rank one, exactly as in the proof of Theorem~\ref{thm:cluster-cut}. Averaging over $g$ and expanding both factors in the isotypic bases gives, by the Schur orthogonality relation $\int_G D^\lambda(g)_{cb}D^\lambda(g)_{da}\,dg=\tfrac1{d_\lambda}\delta_{cd}\delta_{ab}$ for the (real, orthogonal) irreducible matrix elements $D^\lambda$, exactly Eq.~\eqref{eq:reduced-element} on each isotypic block, with all cross-$\lambda$ contributions vanishing by the same orthogonality relation applied to inequivalent irreducibles. Equation~\eqref{eq:per-block-bound} then follows from the standard nuclear-norm bound $\norm{AB}_*\le\norm{A}_\fro\norm{B}_\fro$ applied to Eq.~\eqref{eq:reduced-element}, using $\norm{\tilde r_\lambda}_\fro=\norm{r_\lambda}$, $\norm{\tilde v_\lambda}_\fro=\norm{v_\lambda}$.
\end{proof}
\begin{corollary}[Consistency with the cut-separable bound]
\label{cor:consistency}
Under the hypotheses of Proposition~\ref{prop:reduced-formula},
\begin{equation}
\sum_\lambda\dim(V_\lambda)\,\norm{A_\lambda}_*
\;\le\;
\norm{r^{(S)}}\,\norm{v_{S^c}}
\;\le\;
\sqrt{(d_S-1)(d_{S^c}-1)},
\label{eq:cs-recovery}
\end{equation}
the first inequality by Cauchy--Schwarz over $\lambda$ applied to Eq.~\eqref{eq:per-block-bound}, and the second by the correlation-sum identity used in Theorem~\ref{thm:cluster-cut}. By linearity and convexity of the nuclear norm, Eq.~\eqref{eq:cs-recovery} extends to arbitrary $G$-invariant separable states (finite or continuous mixtures of twirled product terms), recovering the bound of Theorem~\ref{thm:cluster-cut} through the block decomposition rather than around it.
\end{corollary}
\begin{remark}[No universal per-block sharpening, except by concentration]
\label{rem:no-universal-sharpening}
The first inequality in Eq.~\eqref{eq:cs-recovery} is generically strict: equality in Cauchy--Schwarz requires $\norm{r_\lambda}\propto\norm{v_\lambda}$ across all $\lambda$, which independently chosen $\rho_S,\sigma_{S^c}$ have no reason to satisfy. A direct numerical check on a twirled random product state gives $\norm{\mathcal M_S(\rho)}_*=0.429$ against the bound $\norm{r^{(S)}}\norm{v_{S^c}}=0.750$ from Eq.~\eqref{eq:cs-recovery} --- a strict, and generic, gap. Since Eq.~\eqref{eq:per-block-bound} bounds $\norm{r_\lambda}$ only by the global $\norm{r^{(S)}}^2\le d_S-1$, with no constraint on how the purity budget distributes across $\lambda$ for a generic $\rho_S$, no universal constant improving on $\sqrt{(d_S-1)(d_{S^c}-1)}$ holds for an individual block $\lambda$ in general.
The exception is the case where $\V_0^{(S)}$ itself carries only a single isotypic component under $G$: then $r_\lambda=r^{(S)}$ trivially, the full purity budget sits in the one available block, and Eq.~\eqref{eq:per-block-bound} becomes
numerically identical to the cut bound in \cite{aschauer2026a}, but now a statement about a matrix of size $\dim M_\lambda^{(S^c)}\times1$ rather than the full target space. This single-isotype case is a genuine sharpening of concentration, not of threshold, and applies whenever $\rho$ is actually $G$-invariant for a group under which the source side is irreducible. It does \emph{not}, however, apply to the two states used to motivate this section: $\GHZ_3$ and the Smolin state carry no continuous collective symmetry (a direct check shows both fail to be invariant already under a one-parameter collective rotation), so the representation-theoretic mechanism above is not the explanation for their observed degeneracies. The correct explanation for those two states, and more generally for any Pauli-diagonal state, is combinatorial rather than representation-theoretic, and is developed next.
\end{remark}
\subsection{Axial $U(1)$ symmetry: an exact worked example}
The isotypic mechanism of Proposition~\ref{prop:block-diagonal} is easiest to see concretely for an abelian symmetry group, where it reduces to an ordinary charge-conservation selection rule. That a global $U(1)$ (or other abelian) symmetry splits the singular values of a bipartite correlation-type matrix into charge sectors is by now a recurring theme in the entanglement-detection literature under the name \emph{symmetry-resolved entanglement}: it has been used to sharpen partial-transpose-moment witnesses \cite{nevenetal2021}, to decompose the singular values of the realignment matrix itself into $U(1)$-charge blocks \cite{rathetal2023}, and, in a very recent and structurally close construction, to build symmetry-resolved bounds on the negativity directly from charge-sector-resolved realignment-matrix singular values \cite{tarabungahaug2025}. That literature block-diagonalizes the reduced state $\rho_A$ itself (or a moment thereof) and works almost exclusively with abelian charges; Proposition~\ref{prop:block-diagonal} instead block-diagonalizes the linear \emph{response map}$\mathcal M_S(\rho)$, and, in Section~\ref{sec:full-su2-symmetry} below, does so for the non-abelian group $SU(2)$ via the Casimir operator rather than a conserved charge. The mechanism below is the same charge-conservation selection rule applied instead to the bigraduated shadow map $\mathcal M_S(\rho)$, worked out completely and in closed form for one illustrative four-qubit family rather than as a general witness-improvement scheme.
\begin{remark}[Abelian symmetries as selection rules]
\label{rem:abelian-selection-rule}
Let $G=U(1)$ act by collective $z$-rotation, $g=\theta\mapsto\bigotimes_{a\in P}R_z^{(a)}(\theta)$ with $R_z(\theta)=e^{-i\theta Z/2}$. Every irreducible representation of $U(1)$ is one-dimensional, so the isotypic decomposition of Proposition~\ref{prop:block-diagonal} coincides exactly with the eigenspace decomposition of the conserved total charge $S_z^{\mathrm{tot}}$. Writing $E_\pm=(X\pm iY)/\sqrt2$ for the weight-$(\pm1)$ combinations and $E_0=Z$ for the weight-$0$ direction, so that $\mathrm{Ad}_{R_z(\theta)}E_\pm=e^{\pm i\theta}E_\pm$ and $\mathrm{Ad}_{R_z(\theta)}E_0=E_0$, $G$-invariance of $\rho$ forces
where $q_S,q_T$ are the total weights of $\sigma_{i_S}$, $\sigma_{i_T}$ in the $\{E_+,E_0,E_-\}$ basis. Hence a source block of total charge $q_S$ couples \emph{only} to a target block of charge $q_T=-q_S$, not $q_T=q_S$: this is the familiar rule that a correlation function survives only between charge-conjugate directions (e.g.\ $\langle S^+S^-\rangle$ need not vanish, $\langle S^+S^+\rangle$ must). For a non-abelian group such as $SU(2)$ this simple picture does \emph{not} extend by using a Cartan generator (e.g.\ $J_z$) in place of the Casimir: diagonalizing $J_z$ only refines each isotypic block further, by magnetic quantum number, and neither reproduces the isotypic block structure of Proposition~\ref{prop:block-diagonal} nor explains why the reduced block $A_\lambda$ of Proposition~\ref{prop:reduced-formula} is independent of that quantum number --- that collapse is a genuine consequence of full non-abelian equivariance, not of commuting with a single conserved charge. The abelian case is special precisely because weight and isotype coincide there.
\end{remark}
To see this mechanism at work on a state that is not simultaneously covered by the stabilizer mechanism of Lemma~\ref{lem:stabilizer-degeneracy}, consider four qubits $A,B,C,D$ and
with $S=\{A,B\}$, $S^c=\{C,D\}$. Both computational-basis terms have Hamming weight $2$, so $\rho_\varphi$ commutes with the collective $U(1)_z$ generated by $R_z(\theta)^{\otimes4}$ for every $\varphi$, but for generic $\varphi$ it is invariant under no larger continuous collective symmetry. A direct computation gives
\begin{align*}
\tr(\rho_\varphi\,X^{\otimes4})&=\sin(2\varphi),
&
\tr(\rho_\varphi\,X_AX_BY_CY_D)&=-\sin(2\varphi),
\\
\tr(\rho_\varphi\,X_AY_BY_CX_D)&=\sin(2\varphi),
&
\tr(\rho_\varphi\,Z^{\otimes4})&=1,
\end{align*}
with all remaining nonzero-weight correlators vanishing identically; the first three are $\pm1$ only at the isolated points $\varphi=\pi/4\ (\mathrm{mod}\ \pi/2)$, so $\rho_\varphi$ is generically not Pauli-flat and Lemma~\ref{lem:stabilizer-degeneracy} does not apply to it.
For each two-qubit cluster $S$ and $S^c$, the local weight decomposition $\{-1,0,+1\}\otimes\{-1,0,+1\}$ gives total-charge sectors $q=-2,-1,0,1,2$ of dimension $1,2,3,2,1$ respectively (the coefficients of $(x^{-1}+1+x)^2$), on both the source and the target side. By Remark~\ref{rem:abelian-selection-rule} the normalized bigraduated map $\mathcal M_{AB}(\rho_\varphi)$ of Eq.~\eqref{eq:bigraduated-map} block-diagonalizes according to $q_S=-q_T$, and an exact computation of every block gives
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),
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}.
Remark~\ref{rem:abelian-selection-rule} already anticipated that the non-abelian case needs the Casimir operator, not a Cartan generator, and stops short of working this out. We do so here, for the strong hypothesis of Section~\ref{sec:mechanisms} specialized further: not merely party-local, but \emph{collective}, i.e.\ the same single-qubit unitary $U_g$ acting on every party of the full system $P$ simultaneously,
The representation-theoretic content of what follows --- Casimir diagonalization in place of group averaging, and Wigner--Eckart reduction of a tensor to a Clebsch--Gordan coefficient times a smaller reduced operator --- is not new machinery on its own. It is standard practice under the name \emph{spin-adapted DMRG} in ab-initio quantum chemistry, where exploiting the non-abelian $SU(2)$ spin symmetry of the electronic Hamiltonian via exactly this Wigner--Eckart factorization \cite{sharmachan2012spinadapted} routinely enables density-matrix-renormalization-group calculations on active spaces with well over a hundred orbitals, e.g.\ CAS$(113,76)$ on GPU hardware \cite{menczer2024quarterpetaflops}. We are not aware of this toolbox having been carried through as systematically for the entanglement-witness setting of this note, where the reduced object of interest is a witness-relevant response map rather than a Hamiltonian or an MPS tensor; the two settings share the group theory but differ in what is being reduced and why. A related but distinct use of collective-unitary symmetry appears in \cite{huberklepmagronvolcic2022werner}, which constructs dimension-free entanglement witnesses for Werner states (invariant under the diagonal conjugate action of the full unitary group, not merely collective single-qubit rotation) via a semidefinite-programming hierarchy; the guiding idea --- exploit a fixed symmetry to obtain a structural rather than dimension-dependent result --- is the same in spirit, but the symmetry group, the object being reduced, and the resulting statement (existence of a witness, versus the block structure of a given response map) are all different.
Since $\mathcal V_0^{(a)}\cong\mathbb R^3$ carries the adjoint representation $\mathrm{Ad}_{U_g}$, which factors through $SO(3)=SU(2)/\{\pm\id\}$, only integer angular momenta $j$ occur throughout this subsection; we write $G=SO(3)$ accordingly. Eq.~\eqref{eq:collective-hypothesis} is a special case of the strong, party-local hypothesis of Section~\ref{sec:mechanisms} (hence also of the weak, cut-factorizing one), so Proposition~\ref{prop:block-diagonal} and Proposition~\ref{prop:sector-symmetry-compatible} both apply to every cut simultaneously, with the \emph{same} group element $g$ on both sides of every cut -- this simultaneity is what Section~\ref{sec:cut-independent-data} below exploits.
\subsubsection{Branching}
For a cluster of $k$ qubits, $\mathcal V_0^{(V)}\cong(\mathbb R^3)^{\otimes k}$ decomposes under $G$ as $\bigoplus_j m_j^{(k)}V_j$. The multiplicities are already implicit in Remark~\ref{rem:abelian-selection-rule}'s weight count $N_M^{(k)}$, now taken for general $k$ rather than only $k=2$.
\begin{proposition}[Branching multiplicities from $U(1)$ weight counts]
\label{prop:branching-su2}
Let $N_M^{(k)}$ be the dimension of the weight-$M$ eigenspace of the collective Cartan generator $J_z^{\mathrm{tot}}$ on $(\mathbb R^3)^{\otimes k}$, i.e.\ the coefficient of $x^M$ in $(x^{-1}+1+x)^k$ (Remark~\ref{rem:abelian-selection-rule}, generalized from $k=2$). Then
\begin{equation}
m_j^{(k)} = N_j^{(k)} - N_{j+1}^{(k)},
\qquad j=0,1,\dots,k,
\label{eq:branching-formula}
\end{equation}
with $N_{k+1}^{(k)}:=0$.
\end{proposition}
\begin{proof}
Each irreducible $V_j$ contributes exactly one basis vector of weight $M$ for every integer $M$ with $\lvert M\rvert\le j$, and none otherwise. Summing over the isotypic decomposition, $N_M^{(k)}=\sum_{j\ge\lvert M\rvert}m_j^{(k)}$ for every $M\ge0$; the claim follows by taking the difference of consecutive $M$.
\end{proof}
For $k=3$ (the source cluster size used throughout the worked example below), the trinomial coefficients of $(x^{-1}+1+x)^3$ give $N_0^{(3)}=7$, $N_1^{(3)}=6$, $N_2^{(3)}=3$, $N_3^{(3)}=1$, hence
matching the dimension count $\sum_j(2j+1)m_j^{(3)}=1+9+10+7=27=3^3$. This is the first case with a nontrivial multiplicity ($m_1^{(3)}=3$), which is why the worked example below is built on three-qubit clusters.
\subsubsection{The reduced block, without Clebsch--Gordan coefficients}
\begin{corollary}[Isotypic norm without an explicit coupled basis]
\label{cor:projector-norm-formula}
Under the collective hypothesis~\eqref{eq:collective-hypothesis}, fix a cut $S\mid S^c$ and sectors $V\subseteq S$, $T\subseteq S^c$, and let $P_j^{(V)}$, $P_j^{(T)}$ denote the orthogonal projectors onto the spin-$j$ isotypic components of $\mathcal V_V^{(S)}\cong(\mathbb R^3)^{\otimes\lvert V\rvert}$, $\mathcal V_T^{(S^c)}\cong(\mathbb R^3)^{\otimes\lvert T\rvert}$. Then, writing $M:=M_{V\to T}(\rho)$,
with all three statements exact, not merely bounds.
\end{corollary}
\begin{proof}
By Proposition~\ref{prop:block-diagonal}, $M$ is already block diagonal across isotypes with $M|_j=P_j^{(T)}MP_j^{(V)}=A_j\otimes\id_{2j+1}$ and $M|_j P_{j'}^{(V)}=0$ for $j'\ne j$; since the $P_j^{(V)}$ (resp.\ $P_j^{(T)}$) are mutually orthogonal projectors summing to the identity, $M=\bigoplus_jM|_j$ is an honest orthogonal direct sum on \emph{both} sides, and the singular values of a matrix with orthogonally supported blocks are exactly the union of the blocks' singular values. The nuclear norm of $A_j\otimes\id_{2j+1}$ is $(2j+1)\|A_j\|_*$, since each singular value of $A_j$ is repeated $2j+1$ times.
\end{proof}
Eq.~\eqref{eq:projector-norm-formula} means $\|A_j\|_*$ -- and hence the sector-resolved witness $(2j+1)\|A_j\|_*>1$ of Corollary~\ref{cor:sub-block} applied to $\Pi_{\mathcal V}=P_j^{(V)}$, $\Pi_{\mathcal T}=P_j^{(T)}$ -- is computable from the isotypic projectors \emph{alone}, without ever constructing the coupled (Clebsch--Gordan) basis $\ket{j,m;\mu}$ that Eq.~\eqref{eq:wigner-eckart} presupposes. Only the finer question of \emph{which direction} in the multiplicity space $M_j^{(V)}$ carries the signal -- the "diagnostic resolution" motivation of Section~\ref{sec:introduction} -- needs the coupled basis and hence genuine Clebsch--Gordan data.
\begin{remark}[Exact projectors via the Casimir, not group averaging]
\label{rem:casimir-projectors}
The projectors $P_j$ in Corollary~\ref{cor:projector-norm-formula} can be obtained without any integration over $G$. On $(\mathbb R^3)^{\otimes k}$, let $J_a^{\mathrm{tot}}:=\sum_{l=1}^k\id^{\otimes(l-1)}\otimes J_a\otimes\id^{\otimes(k-l)}$ for $a\in\{x,y,z\}$, where $J_a$ is the spin-$1$ generator $(J_a)_{bc}=-i\epsilon_{abc}$ on a single leg, and set $J^2_{\mathrm{tot}}:=\sum_aJ_a^{\mathrm{tot}}J_a^{\mathrm{tot}}$. Since $J^2_{\mathrm{tot}}$ is Hermitian and its eigenvalues on the isotypic component $V_j$ are exactly $j(j+1)$, the eigenspace projectors of $J^2_{\mathrm{tot}}$\emph{are} the $P_j$, obtainable by a single Hermitian eigendecomposition of a $3^k\times3^k$ matrix -- exact up to ordinary floating-point roundoff, with no Monte-Carlo integration over $SO(3)$ required. This is the construction used in \texttt{scripts/exact\_casimir\_projectors.py}; an earlier Monte-Carlo character-averaging version (integrating $\chi_j(g)\,D(g)$ against Haar measure) was used during development and is superseded by this exact construction throughout.
\end{remark}
\subsubsection{Worked example: a six-qubit singlet network}
\label{sec:singlet-network-example}
Fix six qubits $A,B,C,D,E,F$ and the cut $S=\{A,B,C\}\mid S^c=\{D,E,F\}$, together with the finer cut $S'=\{A,B\}\mid S'^c=\{C,D,E,F\}$. Let
with $\ket{\psi^-}=(\ket{01}-\ket{10})/\sqrt2$, two different perfect matchings of the six qubits into singlets, each straddling both cuts. Both states are pure singlets of the \emph{full} collective rotation $U_g^{\otimes6}$ -- each factor is individually annihilated by $\sum_aJ_a^{\mathrm{tot}}$ -- so every state in $\mathrm{span}\{\ket{\psi_1},\ket{\psi_2}\}$ satisfies the collective hypothesis~\eqref{eq:collective-hypothesis} for the \emph{same}$G$, on \emph{both} cuts simultaneously.
\begin{example}[Aligned singlets: an exactly solvable check]
\label{ex:aligned-singlets}
For $\rho_1=\ketbra{\psi_1}{\psi_1}$, the standard singlet correlator $\bra{\psi^-}\sigma_i\otimes\sigma_j\ket{\psi^-}=-\delta_{ij}$ and independence of the three pairs give, for all $i_A,\dots,i_F\in\{x,y,z\}$,
At the cut $S\mid S^c$, reshaping Eq.~\eqref{eq:aligned-singlet-tensor} into the $27\times27$ matrix $M_{S\to S^c}(\rho_1)$ (rows $(i_A,i_B,i_C)$, columns $(i_D,i_E,i_F)$) gives exactly $M_{S\to S^c}(\rho_1)=-\id_{27}$. A scalar multiple of the identity is scalar in \emph{every} orthonormal basis, so Corollary~\ref{cor:projector-norm-formula} gives $A_j=-\id_{m_j^{(3)}}$ with no computation beyond Proposition~\ref{prop:branching-su2}: $\|A_j\|_*=m_j^{(3)}$, i.e.\ $1,3,2,1$ for $j=0,1,2,3$, and $\sum_j(2j+1)\|A_j\|_*=27=\|M_{S\to S^c}(\rho_1)\|_*$.
At the finer cut $S'\mid S'^c$, the same tensor~\eqref{eq:aligned-singlet-tensor}, reshaped to $9\times81$, has exactly $9$ nonzero entries per row (fixed $i_A,i_B$: the entries $(i_C,i_D,i_E,i_F)=(t,i_A,i_B,t)$ for $t\in\{x,y,z\}$), with pairwise disjoint column supports across rows, so $M_{S'\to S'^c}(\rho_1)$ has $9$ equal singular values $\sqrt3$. Here $M_{S'\to S'^c}(\rho_1)M_{S'\to S'^c}(\rho_1)^T=3\id_9$ exactly, so \emph{every} orthonormal sub-basis of the domain -- in particular each isotypic sector $j=0,1,2$ of the two-qubit source $\{A,B\}$ -- sees the same image norm $\sqrt3$; the resulting equality $\|A_0\|_*\approx\|A_1\|_*\approx\|A_2\|_*\approx\sqrt3$ is therefore a linear-algebra consequence of this isometry property rather than a nontrivial fact about $SO(3)$.
\end{example}
\begin{example}[A superposition: genuine multiplicity structure and a coherence term]
\label{ex:singlet-superposition}
Let $\ket{\Xi(\alpha)}:=(\cos\alpha\ket{\psi_1}+\sin\alpha\ket{\psi_2})/N(\alpha)$, still collectively invariant for every $\alpha$ since $\mathrm{span}\{\ket{\psi_1},\ket{\psi_2}\}$ is a $G$-fixed subspace. Because $\rho_\Xi=\ketbra{\Xi}{\Xi}$ is \emph{not} an affine combination of $\rho_1,\rho_2$ for a coherent (rather than incoherent) superposition, the correlation tensor of $\rho_\Xi$ picks up a coherence term absent from Example~\ref{ex:aligned-singlets}:
where $T_a(\vec\imath):=\bra{\psi_a}\sigma_{\vec\imath}\ket{\psi_a}$ ($a=1,2$; $T_1$ is Eq.~\eqref{eq:aligned-singlet-tensor}) and $C_{12}(\vec\imath):=\bra{\psi_1}\sigma_{\vec\imath}\ket{\psi_2}+\bra{\psi_2}\sigma_{\vec\imath}\ket{\psi_1}$ is a third, fixed (state-family-, not $\alpha$-dependent) real tensor. For this family $\braket{\psi_1}{\psi_2}=1/4$ exactly. Table~\ref{tab:singlet-example} reports $\|A_j\|_*$ at $\alpha=\pi/5$ for both cuts, obtained via Corollary~\ref{cor:projector-norm-formula} with the exact Casimir projectors of Remark~\ref{rem:casimir-projectors}; unlike Example~\ref{ex:aligned-singlets}, the values genuinely differ across $j$, and at the cut $S\mid S^c$ the $A_1$ block is a bona fide $3\times3$ matrix (multiplicity $m_1^{(3)}=3$ on both sides) with singular values $(0.6158,0.2171,0.2171)$, not a multiple of the identity.
\caption{$\|A_j\|_*$ at $\alpha=\pi/5$ (Example~\ref{ex:singlet-superposition}), both cuts, from exact Casimir projectors; $\sum_j(2j+1)\|A_j\|_*$ matches $\|M\|_*$ computed directly by brute-force simulation to $10^{-6}$ in both cases (reproduced by \texttt{scripts/apply\_exact\_projectors.py}). No $j=3$ block exists at the $2\mid4$ cut since the two-qubit source has $m_j^{(2)}=0$ for $j>2$.}
\label{tab:singlet-example}
\end{table}
\end{example}
\subsubsection{Cut-independent data: one triple $(T_1,T_2,C_{12})$ for every cut}
\label{sec:cut-independent-data}
The point of Eq.~\eqref{eq:coherence-decomposition} is that $T_1,T_2,C_{12}$ do not depend on $\alpha$\emph{or} on the choice of cut: they are fixed tensors in $(\mathbb R^3)^{\otimes6}$, and every cut's shadow-map block for every $\alpha$ is obtained from them by the fixed linear combination~\eqref{eq:coherence-decomposition} followed by an ordinary reshape, with no further contraction over the full Hilbert space. This was verified directly: brute-force simulation of $\rho_\Xi(\alpha)$ for several $\alpha$, both cuts, matches the prediction from $(T_1,T_2,C_{12})$ to machine precision throughout (\texttt{scripts/recoupling\_check.py}). We record the general statement.
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
$\mathcal C(\rho)(\vec\imath)=\tr(\rho\,\sigma_{\vec\imath})$ 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 $T_{ab}$.
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 $c$,
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.
Take $\ket{\phi_1},\ket{\phi_2}$ as in Eq.~\eqref{eq:singlet-network-states} together with a third perfect matching $\ket{\phi_3}\propto\ket{\psi^-}_{AF}\ket{\psi^-}_{BD}\ket{\psi^-}_{CE}$; all three are collectively invariant, with pairwise overlaps $\braket{\phi_a}{\phi_b}=1/4$ ($a\ne b$) and Gram matrix of condition number $2$ (linearly independent). For a Haar-random Hermitian PSD $c$ of full rank $3$ (not proportional to a rank-$1$ projector, i.e.\ $\rho=\Phi c\Phi^\dagger/\tr(\Phi c\Phi^\dagger)$ a genuinely mixed state with three distinct nonzero eigenvalues), Eq.~\eqref{eq:general-coherence-decomposition} -- built once from the $9=3^2$ tensors $T_{ab}$ -- reproduces the brute-force correlation tensor of $\rho$ to $10^{-16}$, at both the $ABC\mid DEF$ and $AB\mid CDEF$ cuts simultaneously, with no further contraction over the $64$-dimensional Hilbert space (\texttt{scripts/general\_r\_check.py}). This extends the verification of Section~\ref{sec:cut-independent-data} from a pure two-state superposition to a genuinely mixed three-state case, as Proposition~\ref{prop:coherence-templates} and Corollary~\ref{cor:real-tensor-count} require.
\begin{remark}[What this does and does not establish]
\label{rem:recoupling-scope}
Proposition~\ref{prop:coherence-templates} is a bookkeeping consequence of the linearity already used throughout this note (cf.\ the unfolding proposition and the "one rank-one fact, inherited everywhere" theorem of \cite{aschauer2026a}); its content is operational rather than a new inequality: once the $r^2$ template tensors are computed -- from simulation, or in principle extracted from permutationally/collectively-adapted state tomography -- every cut, every mixing angle, and every isotypic block $\|A_j\|_*$ of Corollary~\ref{cor:projector-norm-formula} for the resulting state family is available without revisiting the full $\bigl(\mathbb R^3\bigr)^{\otimes n}$-sized contraction again.
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.
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
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}.)
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; 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.
for a single scalar $c(p,y,j)$. The identical argument, applied with the $C$-leg left unpaired, gives the reduced three-index object $R(m_p,m_C,m_j):=\widehat T(u^{AB}_{p,m_p}\otimes e_{m_C},\,v_{y,j,m_j})$ in terms of the \emph{same} scalar $c(p,y,j)$: since $R$, viewed as pairing the $j$-isotype of $V_p\otimes V_1$ against $V_j$, is subject to the same metric-form constraint,
(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; 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
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; 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
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 $10^{-15}$.
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.
The two reduction mechanisms developed in this note -- the $S_m$-permutation collapse of Proposition~\ref{prop:multinomial-collapse} and the $SO(3)$ branching of this subsection -- combine multiplicatively rather than merely coexisting, and the combination is worth making explicit because it is genuinely stronger than either alone.
\begin{proposition}[Harmonic decomposition of symmetric tensors; classical]
\label{prop:harmonic-decomposition}
As an $SO(3)$-representation, $\mathrm{Sym}^m(\mathbb R^3)$ decomposes multiplicity-free,
where $\mathcal H_j$ denotes the space of degree-$j$ solid harmonics (equivalently, the traceless part of $\mathrm{Sym}^j(\mathbb R^3)$). This is the standard trace-decomposition of symmetric tensors underlying the multipole expansion; see e.g.\ Fulton--Harris.
\end{proposition}
Consequently, for a source cluster $S$ of size $m$ that is invariant under \emph{both}$S_m$ (permutations of its own $m$ parties) \emph{and} the collective hypothesis~\eqref{eq:collective-hypothesis}, the source sector $\mathcal V_S^{(S)}$ collapses in two independent, compatible steps rather than one: first from $3^m$ to $\binom{m+2}{2}$ via Proposition~\ref{prop:multinomial-collapse} (a polynomial, but quadratic, reduction), and then, \emph{within} that $\binom{m+2}2$-dimensional space, the isotypic multiplicities collapse to exactly $1$ for each of the $O(m)$ surviving values $j=m,m-2,\dots$ -- dramatically finer than the generic branching multiplicities $m_j^{(m)}$ of Proposition~\ref{prop:branching-su2}, which grow with $m$ (e.g.\ $m_1^{(3)}=3$ unrestricted, versus multiplicity exactly $1$ within $\mathrm{Sym}^3$). Computationally, this second step costs one further Hermitian eigendecomposition of the $J^2$ Casimir restricted to the already-small $\binom{m+2}2$-dimensional space -- i.e.\ diagonalizing a matrix of side length $\binom{m+2}2$, not $3^m$ -- after which every surviving channel carries a genuine \emph{scalar} reduced matrix element $A_j$ (no residual multiplicity, hence no Clebsch--Gordan bookkeeping of the kind needed in Section~\ref{sec:six-j-recoupling}).
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 $M_{S\to S^c}(\ketbra{Q_3}{Q_3})$ 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
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}):
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.
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 $I\mapsto(0,0)$, $X\mapsto(1,0)$, $Y\mapsto(1,1)$, $Z\mapsto(0,1)$, 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
$\operatorname{Tr}[\rho_H\,\sigma_{\vec i}]= s(\vec i)$ 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).
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.
\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 $s$.
\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
\[
\widetilde{\mathcal M}_S(\rho_H) \text{ has exactly } |\operatorname{im}\varphi|-1 \text{ equal nonzero singular values, each } =\sqrt{|\ker\varphi|}.
\]
After the normalization of Eq.~\eqref{eq:bigraduated-map}, the singular values of $\mathcal M_S(\rho_H)$ are therefore
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}.
The Lemma applies uniformly to pure stabilizer states ($|H|=2^n$, including all graph states and $\GHZ_n$) and to uniform mixtures over a stabilizer code space with $|H|<2^n$ (including the Smolin state, $H=\{IIII,XXXX,YYYY,ZZZZ\}$). Injectivity of $\psi$ holds automatically whenever $H$ contains no element supported entirely on $S$ --- checkable by inspection of the generators, without any Lie-group input.
For $\GHZ_3$ with $S=\{1\}$: $H=\{III,XXX,ZZI,ZIZ,IZZ,YYX,YXY,XYY\}$ (up to signs), $|\ker\varphi|=2$, $|\operatorname{im}\varphi|=4$, $(d_S-1)(d_{S^c}-1)=1\times3=3$, giving $3$ singular values equal to $\sqrt{2/3}$ --- exactly the value reported in \cite{aschauer2026a}.
For the Smolin state, every $1\mid3$ and $2\mid2$ cut has $\varphi$ bijective ($|\ker\varphi|=1$), so $|\operatorname{im}\varphi|-1=3$ in both cases --- bijectivity of $\varphi$ holds for \emph{any} nonempty proper subset $S$ of the four legs given this particular $H$. This gives $3$ equal singular values throughout, with value
\[
\sqrt{\frac{1}{(d_S-1)(d_{S^c}-1)}}
=
\begin{cases}
1/\sqrt7 &\text{for the } 1\mid3 \text{ cut } ((d_S-1)(d_{S^c}-1)=1\times7),\\[2pt]
1/3 &\text{for the } 2\mid2 \text{ cut } ((d_S-1)(d_{S^c}-1)=3\times3),
\end{cases}
\]
matching the results in \cite{aschauer2026a} exactly.
\end{corollary}
\begin{remark}
The $y$-parity grading occasionally useful for real-in-the-computational-basis states is the special case $H=\{I^{\otimes n}\}$ acting trivially --- more precisely, it is not itself an instance of this Lemma but a compatible, coarser $\mathbb Z_2$-grading that commutes with any $H$-decomposition and can be applied on top of it without modification.
\end{remark}
\begin{remark}[Two complementary mechanisms]
\label{rem:two-mechanisms}
Proposition~\ref{prop:block-diagonal} and Lemma~\ref{lem:stabilizer-degeneracy} are complementary, not competing, and apply under disjoint hypotheses. The representation-theoretic mechanism applies whenever $\rho$ is genuinely $G$-invariant under some compact group $G$ acting by local unitaries and preserving the cut, regardless of whether $\rho$ is Pauli-diagonal; it says nothing about states, such as a generic finite-group-symmetric state built from a permutation representation, that are not Pauli-diagonal. The stabilizer mechanism applies whenever $\rho$ is (a uniform mixture over) a stabilizer code state, regardless of whether it possesses any continuous symmetry at all --- as is the case for $\GHZ_n$, the Smolin state, and every graph state used elsewhere in this note, none of which is invariant under a nontrivial continuous collective symmetry. In the (comparatively narrow) overlap where a state is both $G$-invariant for some continuous $G$ and Pauli-diagonal, both mechanisms apply and constrain the same block structure from different directions; outside that overlap, exactly one of the two is available, and it is this Lemma, not Proposition~\ref{prop:block-diagonal}, that accounts for every numerically observed degeneracy reported so far in this note.
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
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.
