Canonical identity

$V=V_2^{\otimes3}$, $\dim V=125$.

$C=(J_x^{\mathrm{tot}})^2+(J_y^{\mathrm{tot}})^2+(J_z^{\mathrm{tot}})^2$.

$K=(C-6I)(C-30I)$.

$E_{47}=\ker K$, $\dim E_{47}=47$.

$\Omega_c=\operatorname{Tr}(P_{47})/125=47/125=0.376$.

Casimir spectrum

$\sigma(C)=\{0,2,6,12,20,30,42\}$.

Verified consequences

1. Closed quantum dynamics: $[H,P]=0$ and $U(t)E_{47}\subseteq E_{47}$ for $H=f(C)$.
2. Quantum filtering: $M_0^\dagger M_0+M_1^\dagger M_1=I$ and $M_0P=P$.
3. Open systems: $\mathcal D_K(\rho)=0$ for $\operatorname{supp}(\rho)\subseteq E_{47}$.
4. Quantum error correction: $PE_a^\dagger E_bP=\alpha_{ab}P$ for $E_a\in\operatorname{span}\{I,K,K^2\}$.
5. Statistical mechanics: $\rho_\beta=e^{-\beta K^2}/Z\to P/47$ as $\beta\to\infty$.
6. Quantum statistics: $\mathbb E_{\mathrm{Haar}}\langle\psi|P|\psi\rangle=47/125$.
7. Lattice field dynamics: $\ker(L_{\mathrm{FCC}}\otimes I+I\otimes K^2)=\operatorname{span}\{\mathbf1\}\otimes E_{47}$.
8. Diffusion: $\partial_t\rho=-D\rho$ converges with gap $\min(\lambda_2(L),\gamma_K)$.
9. Wave mechanics: $\partial_t^2u+Du=0$ has exactly $47$ zero-frequency modes.
10. Information geometry: $\operatorname{Tr}(P)/\operatorname{Tr}(I)=47/125$.

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