diff --git a/paper/symmetric_shadow_maps_formal.tex b/paper/symmetric_shadow_maps_formal.tex
index a4305b8..298b896 100644
--- a/paper/symmetric_shadow_maps_formal.tex
+++ b/paper/symmetric_shadow_maps_formal.tex
@@ -16,6 +16,7 @@
 \newtheorem{definition}{Definition}
 \newtheorem{corollary}{Corollary}
 \newtheorem{remark}{Remark}
+\newtheorem{lemma}{Lemma}
 
 \newcommand{\tr}{\operatorname{tr}}
 \newcommand{\id}{\mathbbm{1}}
@@ -642,6 +643,158 @@ So the same state sits exactly at the boundary for every $2\mid2$ cut while clea
 %% \end{equation*}
 %% This is consistent with the Smolin state being separable across every $2\mid2$ cut while violating the $1\mid3$ bound at $3/\sqrt7\approx1.134$, computed below in Section~\ref{sec:qubit-numerics}. The refinement from singleton to cluster sources costs nothing in the proof yet correctly distinguishes the two cut types, where the singleton construction of Definition~\ref{def:combined-shadow} cannot even pose the $2\mid2$ question.
 
+
+
+\section{Symmetry-adapted block decomposition of the shadow map}
+\label{sec:symmetry-blocks}
+
+The constructions of Sections~\ref{sec:response-maps}--\ref{sec:multiparty-sources} treat the sector grading $V\subseteq S$, $T\subseteq S^c$ as the only available organizing structure on either side of a cut. When $\rho$ carries an additional symmetry --- exactly, or after an appropriate twirl --- this grading can be refined further, along representation-theoretic rather than combinatorial lines. The refinement is worth having for four distinct reasons, which we state before developing the formal statements, since they motivate different parts of what follows and are not equally strong.
+
+\begin{itemize}
+\item \emph{Explanatory power.} Several numerical facts already reported in this note --- the triple degeneracy of the singular values of $\mathcal M_a$ for $\GHZ_3$ (Section~\ref{sec:qubit-numerics}), the triple degeneracy at the Smolin state under both the $1\mid3$ and $2\mid2$ cuts (Section~\ref{sec:multiparty-sources}), and the rigid common value across all $38$ four-qubit graph states --- have so far been recorded as numerical observations. The block-diagonality and stabilizer-support results below show that such degeneracies are not coincidental: they are forced exactly, by two complementary mechanisms depending on whether $\rho$ is symmetry-invariant in a representation-theoretic sense or Pauli-diagonal in a stabilizer sense. This converts a family of separately verified numerical facts into structural statements.
+\item \emph{Diagnostic resolution.} The sub-block witnesses of Corollary~\ref{cor:sub-block} resolve a violation only down to the level of \emph{which parties} are involved ($V,T$). A representation-theoretic decomposition, where applicable, resolves it further, down to \emph{which symmetry channel} within a fixed $(V,T)$ sector is responsible --- for instance, whether a two-party source correlation block is carrying its signal in a totally symmetric or in an antisymmetric combination of its constituents. This is invisible to the party-indexed grading alone.
+\item \emph{Computational cost.} For a source or target cluster respecting a symmetry group $G$, an isotypic decomposition replaces one singular value decomposition on the full sector space by several independent, much smaller singular value decompositions on the multiplicity spaces $M_\lambda$, whose dimensions grow far more slowly than the ambient sector dimension as the cluster size increases. This is the natural computationally tractable foothold for the symmetric sub-family of the higher-order tensor construction discussed in the closing remark of Section~\ref{sec:tensor-viewpoint}.
+\item \emph{A one-sided extension via twirling.} If $\rho$ itself lacks the relevant symmetry but a twirl $T_G(\rho)$ is cheap to evaluate, entanglement detected on $T_G(\rho)$ certifies entanglement of $\rho$ (twirling by local unitaries preserves separability), so a block-diagonal criterion can serve as an inexpensive pre-test, with the standard one-sided caveat that a negative result on $T_G(\rho)$ is uninformative about $\rho$.
+\end{itemize}
+
+What this refinement does \emph{not} deliver is a numerically sharper detection threshold: Remark~\ref{rem:no-universal-sharpening} below shows that no universal, state-independent improvement over Theorems~\ref{thm:cut-bound} and \ref{thm:cluster-cut} exists at the level of an individual representation-theoretic block, except in the single-isotype case where the improvement is one of concentration rather than of threshold value.
+
+\subsection*{Setup}
+
+Let $G$ be a compact group acting on the system by local unitaries, $g\mapsto \bigotimes_{a\in P}U_g^{(a)}$, and suppose $\rho$ is $G$-invariant: $(\bigotimes_a U_g^{(a)})\,\rho\,(\bigotimes_a U_g^{(a)})^\dagger=\rho$ for all $g\in G$. Fix a cut $S\mid S^c$ preserved by $G$ as a set partition, so that $G$ acts on $\V_0^{(S)}$ and on $\V_0^{(S^c)}$ separately, via the adjoint representations $\mathrm{Ad}^{(S)}_g$, $\mathrm{Ad}^{(S^c)}_g$. Decompose both traceless spaces into isotypic components,
+\begin{equation}
+\V_0^{(S)}=\bigoplus_\lambda M_\lambda^{(S)}\otimes V_\lambda,
+\qquad
+\V_0^{(S^c)}=\bigoplus_\lambda M_\lambda^{(S^c)}\otimes V_\lambda,
+\label{eq:isotypic-decomp}
+\end{equation}
+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 Sections~\ref{sec:response-maps}--\ref{sec:multiparty-sources}, now taken with respect to a representation-theoretic rather than a combinatorial grading; the two coincide only in special cases.
+
+\begin{proposition}[Exact block-diagonality]
+\label{prop:block-diagonal}
+Under the hypotheses 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,
+\begin{equation*}
+\Pi_\lambda^{(S^c)}\,\widetilde{\mathcal M}_S(\rho)\,\Pi_\mu^{(S)}=0
+\qquad\text{whenever }\lambda\neq\mu,
+\end{equation*}
+exactly, not merely approximately or in a bound. Moreover, on the surviving diagonal blocks,
+\begin{equation}
+\widetilde{\mathcal M}_S(\rho)\big|_\lambda = A_\lambda\otimes\mathrm{id}_{V_\lambda}
+\label{eq:wigner-eckart}
+\end{equation}
+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}
+
+\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
+\begin{equation}
+A_\lambda=\frac{1}{d_\lambda}\,\tilde v_\lambda\,\tilde r_\lambda^{\,T},
+\label{eq:reduced-element}
+\end{equation}
+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,
+\begin{equation}
+\dim(V_\lambda)\,\norm{A_\lambda}_*\;\le\;\norm{r_\lambda}\,\norm{v_\lambda}.
+\label{eq:per-block-bound}
+\end{equation}
+\end{proposition}
+
+\begin{proof}
+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
+\begin{equation*}
+\dim(V_\lambda)\,\norm{A_\lambda}_*\;\le\;\sqrt{(d_S-1)(d_{S^c}-1)},
+\end{equation*}
+numerically identical to Theorem~\ref{thm:cut-bound}, 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*{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 Sections~\ref{sec:multiparty-sources} and~\ref{sec:qubit-numerics}.
+
+\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
+\[
+\rho_H \;=\; \frac{\Pi_H}{\operatorname{rank}\Pi_H}, \qquad \Pi_H=\frac{1}{|H|}\sum_{h\in H} h,
+\]
+the maximally mixed state on the joint $+1$-eigenspace of $H$ (a stabilizer code state; $\rho_H$ is pure iff $|H|=2^n$).
+
+\begin{lemma}[Support of a stabilizer-code state]
+\label{lem:code-support}
+$\operatorname{Tr}[\rho_H\,\sigma_{\vec i}] = \mathbb 1[\vec i\in H]$ for every $\vec i\in\mathbb F_2^{2n}$.
+\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 Section~\ref{sec:qubit-numerics}.
+
+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 Section~\ref{sec:multiparty-sources} and Section~\ref{sec:qubit-numerics} 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.
+\end{remark}
+
 \section{The tensor viewpoint: shadow maps as unfoldings of one full Bloch tensor}
 \label{sec:tensor-viewpoint}