  We have derived a general rule for the appearance of zero-energy modes in
super-chiral defective nanographene. This so-called "super-zero-sum rule"
defines the appearance of zero modes in a new class of materials, which we call
polymerized phenalenyl-tessellation molecules (poly-PTMs). Through theoretical
modeling of the electronic states in these molecular forms, we provide concrete
solutions for achieving the quantum-spin systems needed in quantum-information
devices. The two-dimensional graph of electronic $\pi$-orbitals in the poly-PTM
possesses a number of localized zero modes equivalent to that of vacancies in
PTMs. In addition to the modes confined to each PTM, another type of zero mode
may appear according to the super-zero-sum rule supported by super-chirality.
Since the magnetic interactions among quantum spins in the zero modes are
determined by how they appear (which is governed by the super-zero-sum rule),
our rule is indispensable for designing quantum-information devices using
electron zero modes in poly-aromatic hydrocarbons and defective graphene with
vacancies.
