Proof of the Residue-Form Algebraic Structure of Winding Numbers in the Quantum Hall Effect

Starting from the viewpoint of geometric phase, we prove that the algebraic–topological structure of the integer quantum Hall effect can be regarded as a ``residue-form winding number.'' Specifically, the first Chern number $C_1$ equals the sum of the first-order pole residues of the transition function $\partial_z\!\log g$, i.e.\ the sum of the coefficients of the $z^{-1}$ terms ($a_{-1}$) in the Laurent expansion. This result reveals the complex-analytic essence of quantum-Hall conductance quantization and provides a unified framework that can be generalised to non-Abelian quantum Hall states and higher-dimensional topological phases.