Geometric Dispersion of Gravitational Waves in the Zhang Effective Metric

We investigate the propagation of gravitational waves within the rotational-order geometric unification framework. Using the Zhang Effective Metric, we derive the modified dispersion relation for tensor perturbations. We show that the internal S³ orientation field induces an energy-dependent phase velocity correction, which is governed by the dimensionless combination (kη). While this geometric dispersion is highly suppressed at current LIGO-Virgo-KAGRA (LVK) frequencies and perfectly consistent with O4 dispersive propagation bounds, it becomes significant at ultra-high energies. This provides a distinct falsifiable signature for future high-frequency gravitational wave detectors, without requiring new elementary particles.