Resolving the all-dimension second-family claim of Pál et al.: Section V, paragraph after Eq. (69), p. 023101-8, Physical Review Research 3, 023101 (2021)

We establish the assertion that second-family states exist for all d in Pál, Tóth, Bene and Vértesi, Section V, paragraph after Eq. (69), p. 023101-8 of Physical Review Research 3, 023101 (2021). Cyclic permutation matrices and sum-indexed shield vectors satisfy all of the defining conditions for every integer d >= 2. Their partial-transpose identity gives explicit positive-partial-transpose states on C^(2d) tensor C^(2d), of ranks d^2+2d and 2d^2+d before and after partial transposition, with quantum Fisher information 16 sqrt(d)/(1+sqrt(d)) for the Hamiltonian of that paper. The construction extends to every finite abelian group; its shield projectors have explicit product decompositions. We also obtain the white-noise threshold and the exact dependence on two local Hamiltonian weights, extending the source's formulas to every dimension.