Open-Set Test-Time Adaptation (OSTTA) operates on non-stationary target streams where covariate and semantic shifts coexist. Existing TTA methods often obtain adaptation signals by updating model parameters or maintaining mutable memories, which can be costly and vulnerable to unknown samples. We present TALOR, a lightweight, backward-free rectification module that uses the source linear head as a geometric anchor. The head-induced basis decomposes each normalized feature into high-gain principal coordinates and low-gain tail coordinates. TALOR estimates a soft-routed affine correction in the principal subspace, combining a regime-level principal bias with a tail-conditioned slope that captures sample-level tail-to-principal coupling. Using weighted ridge regression, it subtracts this drift estimate from the principal coordinates, reconstructs the feature and feeds it into the source head. On the ImageNet-C benchmark with six csOOD datasets, TALOR outperforms the next-best UniEnt by 2.2% H-score while running 2.5x faster and using only 12% of its GPU memory; as a post-correction plug-in, it further boosts COME to 68.4% H-score with only 2% additional memory overhead.

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