An exact four-regime frontier for robust relative fairness with selective labels

This technical note determines the minimum additive tolerance needed for a same-world, benchmark-relative false-positive-rate contraction guarantee under selective labels while requiring a fixed positive accuracy gain for each group in every compatible completion. For a four-cell population model with a=7/10, c=1/10, label-revelation probability s=99/100 in the first three cells, and a wholly unlabeled fourth cell, it gives the exact four-piece frontier for every contraction coefficient rho in [0,1] over all group-blind randomized rules q in [0,1]^4, at guaranteed gain gamma=1/25. The attaining rule is unique throughout. The optimal second-cell probability first increases and then decreases together with the fourth-cell action probability as contraction strengthens. Three actual compatible world pairs give sharp lower bounds, and full-box sign conditions establish attainment. The note also proves a two-way normalized-oscillation representation. The statements are conditional population results for the specified rectangular completion family.