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STATUS · 2026-08-19: E0 algebraic theorem + E1 Python validation. Canonical Drive monograph: Finite SU(2) Isotypic Spectral-Selector Compiler and Optimal Kernel-Contraction Theorem.
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For $s\in\tfrac12\mathbb Z_{\ge0}$, $n\in\mathbb N$, and selected total-spin set $S$, define
$$ \mathcal H_{s,n}=V_s^{\otimes n},\qquad C_{s,n}=J_{x,\mathrm{tot}}^2+J_{y,\mathrm{tot}}^2+J_{z,\mathrm{tot}}^2, $$
$$ K_S=\prod_{j\in S}\left(C_{s,n}-j(j+1)I\right),\qquad Q_S=K_S^\dagger K_S. $$
Then
$$ \ker K_S=\ker Q_S=\bigoplus_{J\in S}\mathcal H_J^{\mathrm{iso}}, $$
with orthogonal projector $P_S=\operatorname{proj}(\ker K_S)$.
If $\lambda_-=\lambda_{\min}^+(Q_S)$ and $\lambda_+=\lambda_{\max}(Q_S)$, then
$$ \varepsilon_S^=\frac{2}{\lambda_-+\lambda_+},\qquad \rho_S^=\frac{\lambda_+-\lambda_-}{\lambda_++\lambda_-}, $$
and
$$ (I-\varepsilon_S^*Q_S)^m\to P_S. $$
$$ (s,n,S)=(2,3,\{2,5\})\Rightarrow \dim\mathcal H=125,\quad \operatorname{rank}P_S=47, $$
$$ \Omega_c=47/125=0.376,\qquad \varepsilon^=1/99144,\qquad \rho^=15/17. $$
Five independent SU(2) specializations reproduce the expected representation-theoretic ranks in Python.
Boundary: finite-dimensional SU(2) representation theory and numerical spectral certification. Downstream physical interpretations require separate models.