Sparse-Label Chest Radiograph Triage Under Radiologist Scarcity: A Semi-Supervised Artificial Intelligence Evaluation of Sensitivity Loss Across District Hospital Cohorts

District hospitals in radiologist-scarce health systems increasingly depend on semi-supervised artificial intelligence to triage chest radiographs, yet the sensitivity cost of learning from sparse labels across heterogeneous facility cohorts remains unquantified. the present analysis formalises sparse-label triage as a semi-supervised risk minimisation problem in which the proportion of labelled cases and the between-cohort prevalence shift jointly govern the achievable sensitivity, and it derives a bound on sensitivity loss relative to a fully supervised reference. The analysis is methodological and theoretical: no observational cohort is analysed. Using a declared variance structure with an intraclass correlation of 0.64, a between-cluster standard deviation of 6.0 and a residual standard deviation of 4.5, the operating characteristics of a before-after-control-impact evaluation design were characterised across 2000 replications per design cell for cluster counts of 8, 16, 24 and 32 and effect sizes of 3, 5 and 8 percentage points. The design required 56 clusters to detect a 5 percentage point sensitivity difference at 80 per cent power. Sparse-label triage evaluations systematically understate sensitivity loss when cohort heterogeneity is absorbed into a single pooled estimate, and it specifies the labelling density and cohort count at which the bias becomes material. The contribution is a formal sensitivity-loss bound and a design protocol for evaluating semi-supervised triage under radiologist scarcity.