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原文摘要

Low-label transfer is often decided by a practical question: after a representation looks useful, how many labels are still needed to learn the downstream boundary? Transferability and probe-quality scores characterize global alignment, likelihood, margins, and calibration. Boundary-near target mass complements these diagnostics by measuring the ambiguous regions where scarce labels are most costly. We propose effective boundary complexity, a label-budget diagnostic for frozen representations and linear probes. It measures mass in a thin strip around the learned boundary and normalizes it by class balance, so dense ambiguous regions and predicted-class imbalance are both counted. We prove that, under a smooth representation map, density change and tangential surface change cancel at first order, leaving boundary-normal stretch or compression as the governing factor. This yields a cached-feature estimator that fits a probe on one split and counts independent held-out features near its boundary. Experiments support the diagnostic at three levels. Controlled transformations match the predicted law across 74 maps with 0.38% median relative error. On real frozen features, normal-direction changes move both measured boundary mass and label demand, while tangent controls preserve the first-order coefficient. A differentiable surrogate reduces measured complexity by 16-36% at matched top-1 utility. Finally, CIFAR-100 and Tiny-ImageNet experiments support boundary mass as a second-stage diagnostic for near-tied representations. Five-way experiments show strong OvR-interface backbone-ranking agreement (Spearman 0.872), while acquisition experiments show high correlations between initial boundary mass and realized label budgets across three sampling policies (0.779-0.830). Code is available at https://github.com/qqaqq-code/boundary_mass.