Few-shot label acquisition lacks a label-free signal for when additional labels cease to improve accuracy: existing stopping criteria either require a held-out validation set (violating the few-shot premise) or rely on theoretically ungrounded heuristics, so we introduce the spectral saturation index $S(K)=\mathrm{erank}(\hatΣ_W^{(K)})/K$, the exponential spectral entropy of the pooled within-class covariance normalized by per-class support size $K$, which measures the exploration rate per label and falls below a fixed threshold $τ=0.02$ once the explored spectral subspace saturates and marginal accuracy gains vanish; across 49 real tasks (binary, 5-way, 10-way) and three frozen backbones (PCA-50, CLIP ViT-B/32, DINOv2 ViT-S/14), $S(K)$ correlates strongly with the marginal gain on doubling the support set ($ρ_{\text{pool}}=0.6366$, $p=2.9\times10^{-57}$, cluster-bootstrap 95\% CI $[0.551,0.720]$), a fixed $τ=0.02$ classifies stop/continue decisions with cluster-bootstrap $\mathrm{AUC}=0.787$ (95\% CI $[0.713,0.860]$) with high recall on meaningful gains ($ΔA>1\%$), and a partial correlation controlling for $\log K$ yields $ρ_{\text{partial}}=0.324$ ($p=1.65\times10^{-13}$), confirming $S(K)$ carries spectral information beyond shared $K$-dependence; theory predicts this from first principles, since the population effective rank sets the saturation scale $K_{\text{sat}}\approx\mathrm{erank}(Σ_W)/τ$, $τ=0.02$ sits at the boundary between the first and second descent (Nakkiran et al., 2021), and $O(1/K)$ bias in the sample effective rank explains the small-$K$ hump in $S(K)$; for unregularized linear probes ($C=\infty$), practitioners should halt when $S(K)<0.02$ (PCA-50, hard stop) or monitor $S(K)$ dropping from $\sim0.3\to0.05$ (foundation models, diminishing-returns signal), with computation costing $\sim1$ ms at $d=50$.
Active learning is now standard practice in labeling ecological data, enabling ecologists to quickly process large volumes of field data to understand and monitor natural environments. Current practices evaluate active learning inductively, estimating predictive performance on a held-out test set. We argue that this evaluation is misaligned with most ecological tasks, where the goal is to transductively label an entire pool of data as efficiently as possible. We demonstrate that ignoring the human-in-the-loop underestimates the importance of continuing to label, particularly for classes in the long tail which may be of disproportionate ecological importance (rare species, uncommon behaviors, etc.). Our analysis shows that, for this long tail, the transductive objective shifts importance from prediction to discovery: the true challenge becomes finding "needles in the haystack," examples of rare classes that are embedded within dense regions of abundant classes in the latent geometry, which we quantify with a novel metric of sampling difficulty. Finally, to translate these insights to practical ecological workflows, we propose a conservative hybrid stopping criterion inspired by ecological rarefaction curves, and show that combining predictive performance with discovery criteria reduces premature stopping on long-tailed pools, improving rare-class recovery when discovery, not classification, is the limiting factor.