This paper investigates the generic structure of smooth functions on compact smooth moduli spaces, revealing that, in a typical sense, global sections of finite complexity do not exist. This conclusion carries direct physical significance: in Higgs gauge theory, it corresponds to the non-flatness of the vacuum moduli space SU(2)/U(1) ≃ S2, thereby explaining the appearance of Goldstone modes and topological defects such as monopoles; in low-dimensional quantum systems and anyon statistics, it corresponds to the multivaluedness of the state space and the intrinsic degeneracy of low-energy degrees of freedom. The approach of this work is built entirely on static geometry and topological analysis, relying neither on time nor on dynamical assumptions. It provides a unified, pre-dynamical structural perspective that clarifies the relationship between the local description of effective field theory and the global geometry.
ZiZhu Wang (Fri,) studied this question.