  Utilizing a series of Bott indices formulated through polynomials of position
operators, we establish a comprehensive framework for characterizing
topological zero-energy corner states in systems with chiral symmetry. Our
framework covers systems with arbitrary shape, including topological phases
that are not characterizable by previously proposed invariants such as
multipole moments or multipole chiral numbers. A key feature of our framework
is its ability to capture the real-space pattern of zero-energy corner states.
We provide a rigorous analytical proof of its higher-order correspondence. To
demonstrate the effectiveness of our theory, we examine several model systems
with representative patterns of zero-energy corner states that previous
frameworks fail to classify.
