Evaluation of Concept-Rank Collapse Audits for Computer Vision Agriculture Disease Classifiers Under Low-Label Regimes: An Interpretability Capacity Bound for Smallholder Fungicide Selectivity

Concept-based interpretability methods for plant-disease classifiers are typically evaluated by whether their concept activations correlate with human-annotated labels, but this criterion is silent on a prior question: whether the classifier's internal representation retains enough concept-rank structure for any audit to be informative. the present analysis formalises that prior question as an interpretability capacity bound. A classifier is modelled as a concept-rank operator whose singular values govern how much concept-level signal survives the projection from pixel space to class logits, and a collapse audit is defined as the ratio of retained concept-rank energy to total concept energy. Under a low-label regime, where concept annotations are scarce and the audit must be estimated from few labelled examples, the bound predicts that audit reliability degrades as a function of the concept-rank spectrum and the label budget. The analysis is a theoretical and methodological contribution: it derives the bound, states the conditions under which a collapse audit is identifiable, and characterises the operating characteristics of the audit as a decision procedure under a declared variance structure (between-cluster SD = 6.0, residual SD = 4.5, intraclass correlation = 0.64). Power was estimated across 2000 design cells for cluster counts of 8, 16, 24 and 32 and effect sizes of 3, 5 and 8 percentage points; 52 clusters were required for 5 percentage points at 80% power. The bound reframes concept audits as capacity tests rather than fidelity tests, with consequences for how smallholder fungicide-selectivity decisions should be supported by interpretable disease classifiers.