Fisher Information Velocity: A New Geometric Channel for Precision Glitch Identification in Gravitational-Wave Detectors

Gravitational-wave detectors operate in inherently non-stationary environments, requiring robust detector characterization (DetChar) to distinguish instrumental transients from astrophysical signals. Traditional DetChar frameworks typically rely on morphological classifiers or energy-based projections, such as band-limited root-mean-square (BLRMS) metrics, which can conflate global amplitude scaling with physical reconfigurations of the spectrum. In this work, we introduce Fisher information velocity, a novel geometric channel that models the detector's power spectral density (PSD) as a point on a Riemannian manifold. By tracking the kinematic drift of the noise floor and utilizing exterior algebra to calculate tangent divergence ($\sin \theta$), we mathematically decouple simple energy surges from spectral warps, or differential redistributions of power across frequency bands. Applying this framework via the sgn-drift streaming pipeline to ~40 hours of high-cadence Advanced LIGO O4a data, we evaluate N=282,080 independent manifold velocity samples. High-resolution phase space mapping reveals a bimodal taxonomy of severe instrumental non-stationarity, classifying events into structural pivots (87.2%) and isotropic surges (12.8%). Among co-detected events, the geometric channel achieves higher significance than standard BLRMS monitors in 74% of cases with a median sensitivity ratio of $\Gamma = 1.65$. The two channels detect largely non-overlapping populations, increasing the total anomaly catalog by 87% over BLRMS alone. Systematic validation on 10 confirmed GWTC-4.0 events and ~5,000 simulated injections demonstrates robust insensitivity to astrophysical signals, establishing this geometric channel as a sensitive, complementary, and veto-safe diagnostic for current and next-generation detector networks.