GAPS: Gradient-Aware Adaptation-Gap Scoring for Time-Series Anomaly Detection with Foundation Models

Reconstruction error is the standard signal for unsupervised time-series anomaly detection and has been widely adopted by time-series foundation models (TFMs). It is, however, vulnerable to input noise: the resulting score tracks the noise level itself rather than whether a sample is truly anomalous, and this failure is especially pronounced in domains where the noise envelope itself carries information about the underlying process state. We propose the *adaptation gap* $\text{Diff} = L_{FT} - L_{ZS}$, the difference between fine-tuned and zero-shot reconstruction losses, as a noise-robust complement. We prove that its conditional expectation cancels the input-noise variance and yields a calibration-anchored, population-level mean-separation guarantee. Building on these results we introduce **GAPS** (Gradient-Aware Adaptation-Gap Scoring), which selectively augments $L_{FT}$ with $\text{Diff}$ under a calibration-only routing layer with no label-tuned hyperparameters. Across a synthetic suite, an HRV benchmark whose normal class is intrinsically noisier than its anomaly class, and TSB-AD-U with a pre-registered noise-stress condition, GAPS preserves performance under standard conditions and yields substantial gains under noise stress.