Target-centered gaze interaction requires more than suppressing frame-to-frame fluctuations: target acquisition produces task-aligned changes in gaze-head dynamics, while a gaze trace may retain a persistent target-relative residual direction. We formulate gaze correction as online target-centered gaze-trajectory forecasting and stabilization and introduce GazeFS, which maps a variable-length gaze-head history to the next target-center direction and a short-horizon Search/Focus estimate without target information at inference.

Across 7,960 acquisition episodes from 30 participants, Search-Focus differences remain stable under quality control, onset exclusion, and duration matching. History windows improve phase decoding over the current endpoint, but explicit task progress remains a strong control.

Under the 30-participant, five-fold grouped out-of-fold protocol across three seeds, the reductions relative to raw hold in Focus episode bias, within-episode dispersion, and P90 target error are 0.182 degrees, 0.257 degrees, and 0.400 degrees, with participant-bootstrap 95% confidence intervals excluding zero. Endpoint-free replay from empty history preserves the Focus advantage and yields raw-network phase balanced accuracy/AUPRC of 0.925/0.993; coordinate controls further show that recent history contributes beyond explicit progress metadata.

GazeFS therefore improves Focus target centering and empirical residual contraction while leaving temporal smoothness as a separate objective.