Add scripts for exact Casimir projectors and recoupling proof

- Implemented `2_exact_casimir_projectors.py` to construct exact SO(3) isotypic projectors using the Casimir operator, replacing Monte-Carlo methods.
- Created `3_apply_exact_projectors.py` to apply the exact projectors to example states, calculating ||A_j||_* estimates with improved precision.
- Developed `4_spherical_basis.py` to build a Condon-Shortley-consistent spherical basis for a single spin-1 leg using ladder operators.
- Introduced `6_six_j_recoupling_proof.py` to provide a complete proof of the cut-recoupling formula for full collective SU(2) symmetry, verifying the relationship between reduced blocks A_j^(1) and A_p^(2).
- Added a README file to document the execution order and purpose of each script in the symmetric states project.

paper/shadow_maps_symmetric_states.tex

@@ -747,19 +747,68 @@ $T(\rho)(\vec\imath)=\tr(\rho\,\sigma_{\vec\imath})$ is linear in $\rho$; substi
 \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.
 
-% TODO (deferred to future work, see session notes): the stronger,
-% state-independent question of whether the reduced blocks A_j themselves
-% (not just the raw tensor) transform between two different cuts of the
-% SAME invariant tensor via an explicit, computable recoupling map -- i.e.
-% a Racah/6j-symbol type formula relating A_j at cut S|S^c directly to the
-% A_{j'} at a different cut S'|S'^c, without reshaping the full raw tensor
-% -- is NOT established here. An exact branching-dimension check (m_p^{(4)}
-% = sum over compatible y of m_y^{(3)} for every p=0,...,4) is consistent
-% with such a relation existing, but the explicit coefficients (presumably
-% governed by an SU(2) recoupling/6j-symbol calculation for the six spin-1
-% legs, or by the Brauer-algebra structure of Inv((R^3)^{\otimes 6}) since
-% the example states here are literally perfect-matching invariants) have
-% not been derived or numerically verified. Left for later.
+\end{remark}
+
+\subsubsection{Recoupling the reduced blocks between two cuts}
+\label{sec:six-j-recoupling}
+
+% ============================================================
+% NEW SUBSUBSECTION -- draft. Resolves the open point left in
+% rem:recoupling-scope of an earlier draft: the reduced blocks A_j
+% themselves (not just the raw tensor) DO transform between two cuts of
+% the same invariant tensor via an explicit, closed-form, state-independent
+% map. Derived and verified exactly (sympy, symbolic CG sums; brute-force
+% simulation cross-check at machine precision) in
+% scripts/six_j_recoupling_proof.py. The derivation below is condensed;
+% the script carries every intermediate identity with its own numerical
+% check, in case a step needs to be revisited.
+% ============================================================
+
+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 $(p,y,j)$,
+\begin{equation}
+A_p^{(2)}[y,j] \;=\; -\sqrt{\frac{2j+1}{2p+1}}\;A_j^{(1)}[p,y],
+\label{eq:six-j-recoupling}
+\end{equation}
+independent of $y$.
+\end{theorem}
+
+\begin{proof}[Proof sketch]
+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.
+\begin{equation}
+\widehat T(u_{p,j,m},v_{y,j,m'}) = c(p,y,j)\,(-1)^{j-m}\,\delta_{m,-m'}
+\label{eq:schur-metric-form}
+\end{equation}
+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,
+\begin{equation}
+R(m_p,m_C,m_j) = c(p,y,j)\,(-1)^{j+m_j}\,\langle p,m_p;1,m_C\,|\,j,-m_j\rangle.
+\label{eq:R-metric-form}
+\end{equation}
+(Naively assuming $R\propto\langle p,m_p;1,m_C|j,m_j\rangle$ without the metric/sign flip -- i.e.\ skipping the $C_j$ correction in Step 1 -- reproduces neither the correct selection rule $m_p+m_C+m_j=0$ forced by $\widehat T$'s own weight conservation, nor a $y$-independent final answer; this was the error in an earlier attempt.)
+
+\emph{Step 2 (conjugation of a real-representation-derived multiplet).} For an $n_\ell$-leg multiplet of total spin $J$ built from the single-leg basis, complex conjugation acts by
+\begin{equation}
+\overline{v_{J,m}} = (-1)^{J+n_\ell}\,(-1)^m\,v_{J,-m},
+\label{eq:conjugation-phase}
+\end{equation}
+verified directly for the $3$-leg tree $(DE)F$ ($n_\ell=3$) and the $4$-leg tree $C,(DE)F$ ($n_\ell=4$) in \texttt{scripts/six\_j\_recoupling\_proof.py}, exactly and for every multiplet label. (Eq.~\eqref{eq:conjugation-phase} follows from the standard conjugation identity for Wigner $D$-matrices, $\overline{D^j_{m'm}(g)}=(-1)^{m'-m}D^j_{-m',-m}(g)$, applied inductively through the coupling tree together with Schur's lemma at each step; we verify the closed form directly rather than re-deriving the induction here.)
+
+\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
+\begin{equation}
+\Xi(p,j,m') := \sum_{m_C,m_j}\langle1,m_C;j,m_j\,|\,p,-m'\rangle\,(-1)^{j+m_j}\,\langle p,m';1,m_C\,|\,j,-m_j\rangle,
+\label{eq:xi-sum}
+\end{equation}
+verified exactly (sympy, symbolic Clebsch--Gordan coefficients) to satisfy $(-1)^p(-1)^{m'}\Xi(p,j,m')=\sqrt{(2j+1)/(2p+1)}$ for every valid $(p,j,m')$ with $p,j\le3$ -- in particular independent of $m'$, as it must be, since $A_p^{(2)}$ is $m'$-independent by Corollary~\ref{cor:projector-norm-formula}. Assembling: $A_p^{(2)}[y,j]=\sqrt{(2j+1)/(2p+1)}\,c(p,y,j)=-\sqrt{(2j+1)/(2p+1)}\,A_j^{(1)}[p,y]$, proving Eq.~\eqref{eq:six-j-recoupling}.
+\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.
 \end{remark}
 
 % TODO (outlook, not attempted): combining the S_m-permutation collapse of