Corollary: The three-qubit (1/3,2/3) anchor

For every $\mathsf{S}\in {\mathcal{A}}_3$, measurement of the first qubit of any two-qubit purification of $\mathrm{diag}(1/3,2/3)$ has law $(1/3,2/3)$.

Proof. The state

is invariant under tensor-factor permutations, which act transitively on its three support strings. Its full law is uniform by equivariance and nullity. Its first-qubit marginal is $(1/3,2/3)$, and its reduced state is diagonal with those weights. Pad the two-qubit purification by one zero spectator and apply Lemma lem:locality.