Three Quantum-Geometric Contributions to Cubic Orbital Magnetization

In noncentrosymmetric metals such as $C_{3v}$ topological-insulator surfaces, moir\'e heterobilayers, and zincblende crystals, point-group symmetry can forbid the linear and quadratic electric-field-induced orbital magnetization, leaving the cubic response as the leading signal. Using a Ward-complete finite-momentum cubic Kubo kernel with an antisymmetric linear-in-$q$ projection, we show that the dc response separates into three quantum-geometric channels. These are a mixed electric-magnetic positional-shift quadrupole, a quantum-metric drift term, and an orbital-moment octupole. The three contributions share the same point-group symmetry but differ in their lifetime, frequency, and gate fingerprints. For a warped $C_{3v}$ surface the metric channel obeys the cutoff-independent law $\bar{\chi}_G \propto \mu^{-2}$. We propose third-harmonic magneto-optical Kerr spectroscopy as an experimental route.