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STATUS: PASS 14/14 · E0 exact derivation + E1 independent NumPy float64 reconstruction
The validator reconstructs the spin-2 generators, total Casimir, six tensor permutations, spectral projectors, joint generator, and convergence witness from first principles. No target eigenvectors, ranks, or multiplicities are inserted.
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On $V=V_2^{\otimes 3}$, define
$K=(C-6I)(C-30I)$,
$\Pi_{\mathrm{sym}}=\frac16\sum_{\sigma\in S_3}U_\sigma$,
and
$A_{\mathrm{can}}=K^2+11664(I-\Pi_{\mathrm{sym}})$.
Then
$\ker A_{\mathrm{can}}=\operatorname{im}P_{\mathrm{can}}$,
where
$P_{\mathrm{can}}=P_{47}\Pi_{\mathrm{sym}}=P_6\Pi_{\mathrm{sym}}$ and
$\operatorname{rank}P_{\mathrm{can}}=5$.
The minimax Richardson operator
$\Gamma_{\mathrm{can}}=I-\frac1{99144}A_{\mathrm{can}}$
obeys the attained identity
$\|\Gamma_{\mathrm{can}}^n-P_{\mathrm{can}}\|_2=(15/17)^n$.
| Item | Certified value |
|---|---|
| Ambient dimension | 125 |
| Casimir multiplicities | 1, 9, 25, 28, 27, 22, 13 |
| $\operatorname{rank}\Pi_{\mathrm{sym}}$ | 35 |
| $\operatorname{rank}P_{47}$ | 47 |
| $\operatorname{rank}P_{\mathrm{can}}$ | 5 |
| $\dim\ker A_{\mathrm{can}}$ | 5 |
| $\operatorname{spec}A_{\mathrm{can}}$ | 0:5; 11664:49; 19600:9; 23328:21; 24208:9; 31264:18; 32400:1; 186624:13 |
| $\lambda_{\min}^+$ | 11664 |
| $\lambda_{\max}$ | 186624 |
| $\varepsilon_*$ | 1/99144 |
| $\rho_*$ | 15/17 |
| Identity | Method | Residual | ||
|---|---|---|---|---|
| $P_{47}^2=P_{47}$ | eigenspace projector, Frobenius | 6.5938e-15 | ||
| $P_{\mathrm{can}}^2=P_{\mathrm{can}}$ | eigenspace projector, Frobenius | 4.5446e-15 | ||
| $K P_{\mathrm{can}}=0$ | raw Frobenius | 7.0652e-13 | ||
| $A_{\mathrm{can}}P_{\mathrm{can}}=0$ | raw Frobenius | 1.9834e-10 | ||
| $A_{\mathrm{can}}P_{\mathrm{can}}=0$ | scale-normalized Frobenius | 1.2566e-16 | ||
| $\Gamma_{\mathrm{can}}P_{\mathrm{can}}=P_{\mathrm{can}}$ | Frobenius | 2.0264e-15 | ||
| $\ | Gamma_{mathrm{can}}^{25}-P_{mathrm{can}}\ | _2=(15/17)^{25}$ | equality error | 1.5890e-15 |