Codimension-controlled universality of quantum Fisher information singularities at topological band-touching defects

Topological phase transitions in generic multiband systems are mediated by band-touching defects whose codimension - the number of momentum directions along which the gap closes linearly - varies across universality classes. Although singular behavior of fidelity susceptibilities and quantum Fisher information (QFI) has been computed for specific models, no unifying principle connecting these results has been identified: it has remained unclear whether the controlling variable is spatial dimensionality, band structure, or an intrinsic geometric property of the defect. We resolve this question by showing that the singular contribution to the QFI with respect to the tuning parameter m obeys a universal power-law scaling sim |m|^p-2 for p neq 2, with a logarithmic divergence sim ln(1/|m|) at the marginal codimension p = 2, where p denotes the codimension of the band-touching defect. This exponent is independent of spatial dimensionality, anisotropies, ultraviolet regularization, and additional gapped bands, and is protected by renormalization-group arguments at the linearized fixed point. The result unifies previously isolated observations for SSH chains $p=1$, Chern insulators $p=2$, and Weyl semimetals $p=3$ as instances of a single codimension-dependent universality class, and reveals that only defects with p leq 2 generate divergent information-geometric responses. This establishes a direct and previously missing link between topological classification in momentum space and quantum distinguishability in parameter space, with implications for metrological sensitivity near topological transitions and for the experimental detection of topological criticality via quantum geometric observables.