  Motivated by iodo platinum complexes assembled within a quadratic-prism
lattice, [Pt(C$_2$H$_8$N$_2$)(C$_{10}$H$_8$N$_2$)I]$_4$(NO$_3$)$_8$, we
investigate the ground-state properties of a Peierls-Hubbard four-legged tube.
Making a group-theoretical analysis, we systematically reveal a variety of
valence arrangements, including half-metallic charge-density-wave states.
Quantum and thermal phase competition is numerically demonstrated with
particular emphasis on doping-induced successive insulator-to-metal transitions
with conductivity increasing stepwise.
