Traditional backdoor attacks inject artificial trigger patterns into training data, causing models to misclassify triggered inputs while maintaining normal behavior on clean samples. Since these artificial triggers are absent in clean datasets, standard knowledge distillation (KD) typically eliminates them, leading to the common belief that KD serves as an effective purification defense. In this paper, we propose DAR (Discover-and-Relabel), a support-persistent backdoor method that retains its behavior even after clean-data distillation. DAR identifies naturally abundant patterns in low-dimensional feature subspaces using subspace clustering and applies label-only poisoning to samples that inherently satisfy the discovered rule. At test time, the backdoor is activated by locally modifying the corresponding subspace of the target images. By instantiating spatial and frequency operators, DAR achieves attack accuracy comparable to state-of-the-art backdoor attacks in ImageNet classification and CLIP-based prompt tuning. Notably, DAR keeps attack success above 96% after clean KD and remains defense-resistant in practice. Project page: https://junhoo.me/backdoor

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