Clar's Rule Reveals the Topological Origin of Edge States in π-Conjugated Systems

Edge states in π-conjugated systems govern scattering, magnetism, and fluorescence quenching, yet their origin has remained elusive. We show that Clar's sextet-maximization principle, the resonance rule of aromatic hydrocarbons, captures the molecular and topological origins of edge states. Applied under boundary constraints, Clar's rule predicts radicals at graphene edges and junctions in agreement with Z₂ topological invariants. Extending this framework to finite nanotubes, we derive a chirality-dependent counting rule and introduce explicit "Clar-cell" constructions that eliminate edge states. Excited-state calculations reveal that conventional unit-cell models host dark excitons from edge states, whereas Clar-cell nanotubes remain intrinsically bright even at nanometer lengths. These results resolve the paradox of ultrashort nanotube fluorescence and establish Clar's rule as a molecular framework linking chemistry and topological physics of edge states.