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Curatorial thesis. The Professor’s Cube is the concrete 5×5×5 coordinate realization of the common 125-state tensor carrier. Its product-cycle Laplacian and the E47 Casimir construction have different kernels, but both commute with the same six tensor-factor permutations of $S_3$.
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professors_cube_exhibit_gallery.html
$V=\mathbb C^5\otimes\mathbb C^5\otimes\mathbb C^5\cong\mathbb C^{125}$
$C=(J_1+J_2+J_3)^2,\qquad K=(C-6I)(C-30I),\qquad P_{47}=P_6+P_{30}$
$L=L_{C_5}\oplus L_{C_5}\oplus L_{C_5}$
For every tensor-factor permutation $U_\sigma$, $\sigma\in S_3$,
$[C,U_\sigma]=[L,U_\sigma]=[P_{47},U_\sigma]=0.$
This is the exact bridge: shared carrier + shared factor-permutation symmetry, not equal kernels.
The full carrier decomposes under $S_3$ as
$125=35_{\mathrm{sym}}+10_{\mathrm{anti}}+80_{\mathrm{std}}.$
Inside E47,
$47=5_{\mathrm{sym}}+0_{\mathrm{anti}}+42_{\mathrm{std}}.$
The trivial $S_3$ line in the spin-2 multiplicity space selects the canonical rank-5 descendant
$P_{\mathrm{can}}=P_6P_{\mathrm{sym}},\qquad \operatorname{rank}P_{\mathrm{can}}=5.$
Equivalently, the exhibit resolves the nested geometry
$125\supset47\supset5.$