<aside> ✅
NEW GROUP-THEORETIC CLOSURE · 2026-08-19
Canonical Drive monograph: Professor’s Cube Orbit-Kernel Equivalence and Heat-Projection Theorem.
</aside>
The 15 orthogonal order-four layer rotations generate exactly 10 disjoint orbits on the 125 sites, with sizes
$$ 8,24,12,24,24,6,8,12,6,1, $$
which sum to 125.
For normalized orbit indicators $v_k=|O_k|^{-1/2}\mathbf1_{O_k}$, define
$$ P_{\mathrm{orbit}}=\sum_{k=1}^{10}v_kv_k^T. $$
Then $P_{\mathrm{orbit}}$ is an orthogonal rank-10 projector.
$$ L_{\mathrm{twist}}=\sum_{g=1}^{15}\left[I-\frac12(U_g+U_g^T)\right]\succeq0. $$
Since
$$ x^TL_gx=\frac12\|x-U_gx\|^2, $$
we obtain
$$ \ker L_{\mathrm{twist}}=\bigcap_g\operatorname{Fix}(U_g)=\operatorname{Ran}P_{\mathrm{orbit}}. $$
Hence
$$ 10=\#\text{orbits}=\dim\operatorname{Fix}(G_{\mathrm{Cube}})=\dim\ker L_{\mathrm{twist}}. $$