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Temporal action segmentation (TAS) demands dense temporal supervision, yet most of the annotation cost in untrimmed videos is spent identifying and refining action transitions, where segmentation errors concentrate and small temporal shifts disproportionately degrade segmental metrics. We introduce B-ACT, a clip-budgeted active learning framework that explicitly allocates supervision to these high-leverage boundary regions. B-ACT operates in a hierarchical two-stage loop: (i) it ranks and queries unlabeled videos using predictive uncertainty, and (ii) within each selected video, it detects candidate transitions from the current model predictions and selects the top-$K$ boundaries via a novel boundary score that fuses neighborhood uncertainty, class ambiguity, and temporal predictive dynamics. Importantly, our annotation protocol requests labels for only the boundary frames while still training on boundary-centered clips to exploit temporal context through the model's receptive field. Extensive experiments on GTEA, 50Salads, and Breakfast demonstrate that boundary-centric supervision delivers strong label efficiency and consistently surpasses representative TAS active learning baselines and prior state of the art under sparse budgets, with the largest gains on datasets where boundary placement dominates edit and overlap-based F1 scores.