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,
for a unique linear map $A_\lambda:M_\lambda^{(S)}\to M_\lambda^{(S^c)}$, the reduced shadow map at $\lambda$.
\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.
\end{proof}
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]
\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
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}. 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
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 $\varphi=\pi/4$.
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,
\begin{equation}
g\ \longmapsto\ \bigotimes_{a\in P}U_g,
\qquad
g\in SU(2).
\label{eq:collective-hypothesis}
\end{equation}
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 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
\begin{equation}
T(\rho) = \sum_{a,b} c_{ab}\,T_{ab},
\label{eq:general-coherence-decomposition}
\end{equation}
independent of which cut's matricization 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 $T_{ab}$.
\end{proof}
% TODO (open / not yet carried out): Proposition~\ref{prop:coherence-templates}
% is stated and used above only for r=2 real generators (psi_1, psi_2), where
% the single real cross term C_{12} suffices because the c_{ab} happen to be
% real and symmetric in that example. For general r this requires r(r+1)/2
% independent real tensors (r diagonal T_aa plus the real/imaginary parts of
% the r(r-1)/2 off-diagonal overlaps, or equivalently just track the full
% Hermitian r x r coefficient matrix c_{ab} against the T_{ab} disregarding
% the a<->b symmetry) -- this is mechanical but has not been written out or
% numerically tested here for r>2. Do this before submission if the general
% statement is kept; otherwise restate the Proposition for r=2 only and move
% the general case to a remark.
\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.
\begin{theorem}[Cut recoupling for six spin-1 legs]
\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 $(p,y,j)$,
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.
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).} For an $n_\ell$-leg multiplet of total spin $J$ built from the single-leg basis, complex conjugation acts by
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.)
\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
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}.
\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.
\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}; 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.
\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
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}
$\operatorname{Tr}[\rho_H\,\sigma_{\vec i}]=\mathbb1[\vec i\in H]$ for every $\vec i\in\mathbb F_2^{2n}$.
\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.
\end{proof}
\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
\[
\sqrt{\frac{|\ker\varphi|}{(d_S-1)(d_{S^c}-1)}},
\]
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}.
\end{proof}
\begin{corollary}
\label{cor:stabilizer-examples}
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.
